NOTE: Most people think that the phrase "eternal fire" proves, or at the very least, provides considerable support for, the belief in eternal conscious torment. By the end of this answer, we will see that the term "eternal fire" provides some of the absolute best arguments against the belief in eternal conscious torment, that is, in support of the belief in conditionalism.1 So, let's comprehensively analyze the phrase "eternal fire".
What Really Is "Eternal Fire"?
The phrase "eternal fire" appears only three times in the New Testament (a semantically identical phrase is found once in the Hebrew Bible). It appears in Matthew 18:8, Matthew 25:41, and Jude 7.
Matthew 18:8 If your hand or your foot causes you to sin, cut it off and throw it away. It is better for you to enter life crippled or lame than to have two hands and two feet and be thrown into the eternal fire.
Matthew 25:41 Then He will say to those on His left, ‘Depart from Me, you who are cursed, into the eternal fire prepared for the devil and his angels.
Jude 7 just as Sodom and Gomorrah and the cities around them, having indulged in sexual immorality and having gone after strange flesh, in like manner with them are set forth as an example, undergoing the penalty of eternal fire.
The one often focused on is Matthew 25:41, where Jesus tells the cursed, i.e. the workers of lawlessness (see Matthew 7:23, Luke 13:27), to depart from Him "into the eternal fire prepared for the devil and his angels". The traditional argument based on the phrase "eternal fire" is something like this:
P1: Eternal fire is fire that lasts for eternity.
P2: Fire that lasts for eternity requires fuel that lasts for eternity.
P3: The fuel for the fire is all the wicked, rebellious sinners who are thrown into it.
C: The wicked, rebellious sinners who are thrown into the fire last for eternity (follows deductively from P1, P2, and P3).
I'll grant premise 1 for now. Now about the second premise ... does fire that lasts for eternity require fuel that lasts for eternity? Possibly, but there's nothing in the passage itself that suggests that. Surely, God is capable of creating a fire that lasts for eternity on its own. In fact, according to traditionalism, there are worms that live forever in hell! Actual worms do not live forever, even with an abundance of sustenance (in this case, all the wicked who are thrown into hell). These would have to be special worms created by God for the purpose of eternally consuming the wicked in hell. If God can create worms that last forever, why couldn't He create a fire that lasts forever? The conditionalist could argue that God causes the fire to last forever as a memorial/monument of the permanent destruction of the wicked, and not so that it can torment the wicked for eternity; essentially, the everlasting fire symbolises the everlasting death and destruction of the ungodly (remember, this interpretation presupposes the validity of premise 1 in the argument above). All in all, premise 2 is insubstantial, making this argument against conditionalism a feeble one.
So, what is "eternal fire", and what does it do? Let's start with the first occurrence of the term in the New Testament, namely, Matthew 18:8, and see if there's anything in the context that helps us to figure out what "eternal fire" is.
[Matthew 18:8-9] Now if your hand or your foot causes you to sin, cut it off and cast it from you; it is better for you to enter into life crippled or lame, than having two hands or two feet, to be cast into the eternal fire. 9 And if your eye causes you to sin, gouge it out and cast it from you; it is better for you to enter into life one-eyed, than having two eyes to be cast into the Gehenna of fire. (BLB)
Jesus sets v.8 in parallel with v.9, which means that what is expressed in v.9 is synonymous with what is expressed in v.8 (v.9 reiterates in different words what has already been communicated in v.8). At the start of both verses, Jesus tells us that we must be willing to remove and cast out any body part that causes us to sin,2 because the reward for doing so is great, and the punishment for not doing so is likewise great. What does Jesus say would be the reward for us if we remove any body part that causes us to sin? We will enter into life! He then contrasts that with the punishment for not doing so; we will be cast into the "eternal fire" (v.8), the "Gehenna of fire" (v.9). Immediately, we see that whatever it means to be cast into the "eternal fire", it necessarily entails not entering into life, as being "cast into the eternal fire" is presented as the alternative to "entering into life". Jesus tells us that it is better to enter into life maimed instead of cast into the eternal fire fully intact. Thus, He presents us with two mutually exclusive possibilities:
(1.) Remove any body part that causes us to sin and enter into life.
(2.) Don't remove any body part that causes us to sin, continue to sin, and be cast into the "eternal fire", the "Gehenna of fire".
Therefore, whatever it means to be "cast into the eternal fire", we know that it does not include having life, which means that those who are cast into it must lose their life (which contradicts the notion that the wicked will have their lives forever in torment in the lake of fire).
Of course, we must ask ... what is the "life" in question? Is it physical eternal life (i.e. embodied existence that lasts forever) or "spiritual eternal life" (i.e. "knowing God and having a relationship with Him")? Apart from the fact that physical eternal life is the most natural interpretation of "life" in Matt. 18:8-9, it's also the only one that makes sense. For one, according to believers in "spiritual eternal life", spiritual eternal life is something we gain in this age now. However, "entering into life" in Matt. 18:8 is something that happens in the next age. How do I know? Well, the context is clearly about ultimate fates. Being cast into the eternal fire happens in the next age, not in this one. Likewise, entering into life and into the kingdom of heaven (v.3) is something that happens in the next age. However, gaining "spiritual eternal life" happens now; it's something we attain in this age. And you cannot enter into something if you've had it all along. Therefore, since Jesus cannot be talking about entering into spiritual eternal life, He must be talking about entering into physical eternal life, or as I define in my answer here, embodied existence that lasts for eternity. The word for "life" in Matt. 18:8-9 is ζωή (see Strong's G2222), pronounced zóé. This is the only word Jesus uses to refer to "eternal life" (ζωὴν αἰώνιον[zōēn aiōnion]). Thus, Jesus is telling us that it is better to enter into endless embodied existence maimed than to be cast into the eternal fire. Since being cast into the eternal fire is the alternative to entering into endless embodied existence, it necessarily entails dying (losing your life) at some point (whether at the moment we are cast into it or somewhere down the line). This, in and of itself, disproves the doctrine of eternal conscious torment, which contends that the wicked will receive eternal embodied existence just as the righteous will (it is only a question of where they will be living out their embodied existence. In the case of the righteous, in heaven, while in the case of the unrighteous, in hell).
So far, the term "eternal fire" has provided major support for conditionalism. If "life" (ζωή) is understood in the sense of physical eternal life (embodied existence that lasts for eternity), then being cast into the "eternal fire", which is presented as the alternative to entering into "life" (ζωή), must entail not having eternal embodied existence, which necessitates that those who are cast into it ultimately die, and that contradicts the doctrine of eternal conscious torment/separation. However, there is even more to the term "eternal fire" that works against these doctrines. It's the fact that the "eternal fire" in v.8 is paralleled with the "Gehenna of fire" in v.9 (which means the two are interchangeable). How exactly does this work against ECT? It's because of what "Gehenna" refers to. The word Gehenna in Greek is γεέννῃ, pronounced geennē. It's a transliteration of the Hebrew phrase "Ge-hinnom", which derives its name from the Valley of Ben-Hinnom, also called the Valley of Hinnom, which is the valley right outside of Jerusalem. Contrary to popular belief, it does not refer to a perpetually burning garbage dump in the Valley of Hinnom (see here and here for more). It simply refers to the Valley of Hinnom. So, what is the significance of the Valley of Ben-Hinnom?
In the Hebrew Bible, the Valley of Ben-Hinnom is depicted, not as a place of perpetual torment, nor of separation from the presence of God, but as a place where God's enemies are slain and utterly eliminated. It is where the loathsome and iniquitous Israelites who burned their sons and daughters in honour of false deities were slaughtered by their enemies (see Jeremiah 7:30-34; 19:6-7). The valley is literally called "the Valley of Slaughter" twice! The valley was filled to the brim with their corpses, where they were irresistibly consumed by scavengers until nothing remained. The valley is called the "valley of dead bodies and ashes" (see Jeremiah 31:40). It is where Yahweh waged a raging war with "burning anger" and "consuming fire" against the Israelites' oppressors, the Assyrians (that day is called "the day of great slaughter"); He incinerated their king in Topheth, which is in the Valley of Ben-Hinnom (see Isaiah 30:25-33). It is where God fought a furious battle against the wicked who had rebelled against Him and massacred them; their carcasses were torched and combusted by an inextinguishable blaze, and anything that remained was fully devoured by the detritivores (e.g. worms and maggots), which, in all, would have totally consumed the corpses of those whom God had slain (see Isaiah 66). For an extensive analysis of Gehenna, see my answer here. Needless to say, when Jesus' disciples heard Him use the phrase "Gehenna of fire", they were not thinking of being separated from the presence of God, nor of being unceasingly tormented as living immortals; they were conceiving of perishing violently in inexorable flames! But this is something that the doctrine of eternal conscious torment/separation cannot allow (under such theologies, the wicked must stay embodied throughout eternity; physical death is not an option).
As is evident, using the single occurrence of "eternal fire" in Matthew 18:8, one can make an incredibly robust case against traditionalism. Being cast into the eternal fire is presented as the alternative to entering into life, meaning the punishment of eternal fire must, at the very least, include death (which isn't possible if you're living for eternity in an immortal body). The "eternal fire" is also paralleled with the "Gehenna of fire", and the only thing fire ever did in the Valley of Ben-Hinnom (in the context of God's judgment) was consume the ungodly. It never eternally tormented them (any torment experienced by anyone was temporary and always ended in their demise) or separated them from His presence. Now, what else can we learn about eternal fire? Well, the Hebrew Bible contains a phrase that is semantically identical to "eternal fire", i.e. "everlasting flames/burnings".
[Isaiah 33:10-14] “Now I will arise,” says the LORD. “Now I will lift Myself up. Now I will be exalted. 11 You conceive chaff; you give birth to stubble. Your breath is a fire that will consume you. 12 The peoples will be burned to ashes, like thorns cut down and set ablaze. 13 You who are far off, hear what I have done; you who are near, acknowledge My might.” 14 The sinners in Zion are afraid; trembling grips the ungodly: “Who of us can dwell with a consuming fire? Who of us can dwell with everlasting flames?” (BSB)
Yahweh has proclaimed that He will be exalted and that the peoples will be burned to ashes,3 as "thorns cut down and set ablaze". The sinners and the ungodly are all justifiably petrified, and ask rhetorical questions such as, "who among us can dwell with consuming fire" and "who among us can dwell with everlasting flames/burnings". The answer to these questions is unquestionably "no one". The wicked are like chaff consumed by fire; like thorns cut down and set ablaze, they are reduced to ashes. That's what God's consuming fire does; it's what His everlasting flames do. They don't torment the wicked for eternity or separate them from His presence; they devour the wicked and put them to an end. The sinners in Zion knew this and dreaded it; they were trembling in fear because they knew what God's everlasting flames would do to them, i.e. fully consume them and reduce them to ashes. This is the picture "eternal fire" should bring to our minds ... fire that devours evildoers and reduces them to ashes (cf. Psalm 21:8-9, Psalm 68:2, Malachi 4:1-3, Hebrews 10:27, 2 Peter 2:6).
So far, we've learned a lot about "eternal fire". We know that being cast into the eternal fire is the alternative to living forever (Matthew 18:8). We also know that the wicked being cast into the "eternal fire" at final judgment is the same as them being thrown into the "Gehenna of fire" (Matthew 18:8-9); this expression evokes the Valley of Ben-Hinnom, where the adversaries of God were slaughtered and their corpses incinerated. And finally, we know that it is a fire in which no one can dwell, for it fully devours the ungodly, putting them to an end (Isaiah 33:10-14). All of this is more than enough to disprove the doctrine of eternal conscious torment/separation. However, to drive the final nail into the coffin, we must consider Jude 7.
[Jude 7] just as Sodom and Gomorrah and the cities around them, since they in the same way as these indulged in gross immorality and went after strange flesh, are exhibited as an example in undergoing the punishment of eternal fire. (NASB)
[Jude 7] just as Sodom and Gomorrah and the cities around them, having indulged in sexual immorality and having gone after strange flesh, in like manner with them are set forth as an example, undergoing the penalty of eternal fire. (BLB)
[Jude 7] Even as Sodom and Gomorrah, and the cities about them, having in like manner with these given themselves over to fornication and gone after strange flesh, are set forth as an example, suffering the punishment of eternal fire. (ASV)
[Jude 7] as Sodom and Gomorrah, and the cities around them in a similar manner to these, having given themselves over to sexual immorality and gone after strange flesh, are set forth as an example, suffering the vengeance of eternal fire. (NKJV)
What does Jude mean when he says that the cities of Sodom and Gomorrah "are set forth as an example, undergoing the penalty of eternal fire"? I can think of three important questions that need answering:
(1.) Of whom are the cities of Sodom and Gomorrah set forth as an example?
(2.) In what way are Sodom and Gomorrah, suffering the penalty of eternal fire, exhibited as an example?
(3.) What is the punishment of eternal fire?
Let's start with the first question. I'd say the answer to the first question is pretty obvious ... the cities of Sodom and Gomorrah are set forth as an example of what will happen to the wicked. This is confirmed by Matthew 18:8-9, Matthew 25:41, and the parallel to Jude 7, which is 2 Peter 2:6. In Matthew 18:8-9, the "eternal fire" is set in parallel to the "Gehenna of fire". Most people agree that "Gehenna" refers to the lake of fire or second death, which is the final punishment of the wicked (according to Revelation 20:15, everyone who does not have their name written in the book of life will be thrown into the lake of fire). In Matthew 25:41, Jesus is presented as telling the cursed to depart from Him "into the eternal fire prepared for the devil and his angels". It is clear, then, that the "eternal fire" is the punishment that the unrighteous will undergo. Hence, Sodom and Gomorrah, themselves having suffered the penalty of eternal fire, are on display as an example of what will happen to the wicked. Something remarkably similar is said in 2 Peter 2:6 (which is to be expected, since Jude 7 parallels this passage).
[2 Peter 2:6] if He condemned the cities of Sodom and Gomorrah to destruction, reducing them to ashes as an example of what is coming on the ungodly; (BLB)
The cities of Sodom and Gomorrah, in their condemnation and absolute obliteration, are said here to be an example of what is coming on the ungodly. And, as previously noted, the punishment of eternal fire is what is coming on the ungodly. Thus, Sodom and Gomorrah, themselves having experienced this punishment, serve as an example of what is gonna happen to all of the wicked and corrupt, who will likewise undergo the punishment of eternal fire by being cast into it (in accordance with Matthew 18:8-9, Matthew 25:41, and Revelation 20:15). Hence, the answer to question one is straightforward: The cities of Sodom and Gomorrah, in undergoing the penalty of eternal fire, are exhibited as an example of what will happen to all evildoers at the final judgment.
Now, for the second question: In what way are Sodom and Gomorrah, suffering the penalty of eternal fire, exhibited as an example? The word translated "set forth" or "exhibited" is πρόκειμαι, pronounced prokeimai. According to LSJ and Thayer's Greek Lexicon, the word means "to be set before one, to be placed before the eyes, to lie in sight". So, is Jude telling his audience that they can look before them and see that Sodom and Gomorrah are an example due to undergoing the penalty of eternal fire? Well, not necessarily. The word πρόκειμαι can be used figuratively to denote something cognitively present as opposed to concretely present. In fact, Strong's Concordance actually says that πρόκειμαι can be used figuratively to refer to something that is present in the mind. There is a great example of such a usage of πρόκειμαι by Flavius Josephus in the The Wars of the Jews (you can read it here).
But still, John, it is never dishonorable to repent, and amend what hath been done amiss, even at the last extremity. Thou hast an instance before thee in Jechoniah, the king of the Jews, if thou hast a mind to save the city, 104 who, when the king of Babylon made war against him, did of his own accord go out of this city before it was taken, and did undergo a voluntary captivity with his family, that the sanctuary might not be delivered up to the enemy, and that he might not see the house of God set on fire; (The Wars of the Jews, 6.103-6.104, by Flavius Josephus, translated by William Whiston)
The word translated "before" is the same word translated as "set forth" in Jude 7, namely, πρόκειμαι (you can read the Greek here to confirm what I'm saying). But it's not being used to denote something that is concretely present, that is, something visible before our eyes (even though that is the primary meaning of the word πρόκειμαι). How do I know? Well, because the example laid out before John is Jehoiachin, who lived in the 6th-century BC. His story is recorded in 2 Kings 24:10-13.
[2 Kings 24:10-13] At that time the servants of Nebuchadnezzar the king of Babylon went up to Jerusalem, and the city came under siege. 11 And Nebuchadnezzar the king of Babylon came to the city, while his servants were besieging it. 12 Then Jehoiachin the king of Judah went out to the king of Babylon, he, his mother, his servants, his commanders, and his officials. And the king of Babylon took him prisoner in the eighth year of his reign. 13 He also brought out from there all the treasures of the house of the Lord, and the treasures of the king’s house, and he smashed all the articles of gold that Solomon king of Israel had made in the temple of the Lord, just as the Lord had said.
So, Josephus was saying that Jehoiachin is "set forth" (πρόκειμαι) before John as an example in the pages of scripture. That is precisely what Jude is saying about Sodom and Gomorrah. The cities of Sodom and Gomorrah are set forth before us as an example of what is going to happen to the ungodly through the record of them undergoing the penalty of eternal fire found in the pages of scripture. The record in scripture and the collective memory of what happened to Sodom and Gomorrah are a present reminder of the divine punishment awaiting the wicked at the final judgment. Essentially, the account of the cities of Sodom and Gomorrah "suffering the vengeance of eternal fire" gives us a mental picture of what is going to happen to the unrighteous.
And now for the third question: What is the punishment of eternal fire? We know that the cities of Sodom and Gomorrah underwent the penalty of eternal fire, since Jude tells us as much. So, if we want to understand the nature of the punishment of eternal fire, we must look at the record of Sodom and Gomorrah's destruction found in scripture.
[Genesis 19:23-29] The sun had risen over the earth when Lot came to Zoar. 24 Then the Lord rained brimstone and fire on Sodom and Gomorrah from the Lord out of heaven, 25 and He overthrew those cities, and all the surrounding area, and all the inhabitants of the cities, and what grew on the ground. 26 But Lot’s wife, from behind him, looked back, and she became a pillar of salt. 27 Now Abraham got up early in the morning and went to the place where he had stood before the Lord; 28 and he looked down toward Sodom and Gomorrah, and toward all the land of the surrounding area; and behold, he saw the smoke of the land ascended like the smoke of a furnace. 29 So it came about, when God destroyed the cities of the surrounding area, that God remembered Abraham, and sent Lot out of the midst of the destruction, when He overthrew the cities in which Lot had lived. (NASB)

So this is what the punishment of eternal fire looks like; like a downpour of fire and sulfur that overthrows and destroys entire cities, their surrounding area, and their inhabitants (as well as any flora that grows in the midst of the destruction). The smoke that was a sign of the land's utter destruction was ascending like smoke from a fiery furnace! Eternal fire completely obliterates everything in its path. Indeed, who can dwell in God's everlasting flames? The indisputable answer to this question is no one. The apostle Peter gives an excellent description of the punishment that Sodom and Gomorrah suffered.
[2 Peter 2:6] and if He condemned the cities of Sodom and Gomorrah to destruction by reducing them to ashes, having made them an example of what is coming for the ungodly; (NASB)
The punishment of eternal fire utterly demolishes entire cities; the cities of Sodom and Gomorrah were reduced to ashes! Everyone within them was eradicated off the face of the earth ... they altogether perished. So, to summarize what "eternal fire" is in one succinct expression: fire from God that absolutely decimates and eradicates wicked and corrupt people, reducing them to ashes. This is the punishment of eternal fire.4
There are only two verifiable instances in scripture where people are said to suffer the punishment of eternal fire. The first is in Matthew 25:41, which is a prophetic account depicting Jesus at final judgment telling the workers of iniquity to depart from Him into the eternal fire. Jesus does not tell us what happens to these ones when they go into the eternal fire, so, if that was all we had to go off of, there would be no way to know what is going to happen to those who are cast into the eternal fire at the final judgment, making it, essentially, a free-for-all (something that can be interpreted from a variety of opposing perspectives). Thankfully, that's not all we have. We have one other confirmed record of people undergoing the penalty of eternal fire; they are the cities of Sodom and Gomorrah, according to Jude 7. And this time we explicitly know what happened to those who suffered it. Accordingly, we do not have to guess what it means to suffer the punishment of eternal fire, and interpreting it as whatever fits our theology (e.g. eternal conscious torment or separation from God's presence) is completely unwarranted. In the only other established instance of people suffering the penalty of eternal fire, we know exactly what the eternal fire did; it completely eradicated those who suffered it! Therefore, we know what is going to happen to the workers of iniquity in Matt. 25:41 who sustain the same penalty of eternal fire; they will be exterminated, eliminated, obliterated, just as the cities of Sodom and Gomorrah were when they sustained the very same penalty of eternal fire.
There are absolutely no grounds for saying that something altogether different is going to happen to them, such as that they will be consciously separated from God's presence for eternity or that they will be tormented in immortal bodies in the lake of fire forever. That's completely unjustified. I'll reiterate: We are not in the dark about what eternal fire does. We know precisely what it does, namely, eradicate the wicked and reduce them to ashes. There is no basis whatsoever for saying that the cursed in Matthew 25:41 who will experience the punishment of eternal fire will have something entirely separate done to them, that is, entirely separate from what happened to Sodom and Gomorrah, who are, mind you, the only instance of those who have undergone the punishment of eternal fire where we actually expressly know what happened to those who underwent it. The cursed who depart from Christ into the eternal fire in Matthew 25:41 will be eliminated, extirpated, executed, just as Sodom and Gomorrah were when they underwent the very same retribution. This plain teaching of scripture is, however, irreconcilable with ECT/ECS.
Notes:
1 This is the belief that being granted immortality (the quality of living forever and not being subject to death) by God is contingent upon being a believer in Jesus Christ. Of course, a corollary of this belief is that all unbelievers will eventually perish. It is very similar to annihilationism, but not the exact same. Annihilationism likewise asserts that all unbelievers will perish due to immortality being conditional upon one's belief in Christ, but furthermore, it asserts that unbelievers will permanently cease to exist by having their body and soul destroyed (which does not mean "obliterated from existence", and does not require breaking any fundamental physical laws. See my answer here). Both are incompatible with the traditional doctrine of eternal conscious torment/separation.
2 I do believe that Jesus is employing hyperbole in Matt. 18:8-9, but since that is a separate issue impertinent to our discussion, I will simply take Jesus' words at face value for now.
3 Lit. "burnings of lime". Limestone is easily burnt and reduced to a powdery substance called calcium oxide (also known as quicklime), so the picture here is that the people will be burnt to ashes. The NIV, BSB, CSB, ISV, and NET all translate it as such.
4 It's interesting to note that the fire that rained upon Sodom and Gomorrah is not still burning today, which means that "eternal fire" is not eternal because it lasts for eternity. So, why is it called eternal? Some suggest that the fire is eternal because its consequences, namely, total destruction, are eternal. However, someone recently pointed out to me that this cannot be since, according to Ezekiel 16:53-55, Sodom will be restored to its former state; it will not stay destroyed forever. Therefore, I suggest (I'm not the first) that "eternal fire" is eternal because it comes from an eternal source, namely, God (see Gen. 21:33, Deut. 33:27, Psa. 90:2, Rom. 1:20, Heb. 9:14). In fact, according to Genesis 19:24, the fire and brimstone that rained upon Sodom and Gomorrah came directly from Yahweh out of heaven.