I. Gather the weeds and burn them
Let's start with a face-value reading of Matthew 13:30. Darnel weeds, along with dry grasses and brush (scrub vegetation), are highly combustive. Less combustible materials are more fire-resistant and thus harder to burn up; weeds are not such substances. Like chaff, weeds are easily consumed by fire, and like chaff, weeds do not burn forever, nor for any ostensibly long period of time. According to adherents of the doctrine of eternal conscious torment (ECT), however, the damned in hell are incorruptible, or at the very least not consumed by fire which torments them ... using the image of weeds being burned by fire wouldn't have made much sense if Jesus wanted to communicate the everlasting despair of people who remain alive for eternity.
It would also prove useful to analyze the word Jesus uses to convey what is done to the weeds at the harvest. He uses the compound verb κατακαίω (katakaió), translated as "burn". It is perfective, meaning it stresses the verbal action as a whole, indicating that the action is carried to a fixed end, that is, completed (this is in contrast to the word καίω [kaió], which refers to the setting on fire or burning of a substance without necessarily to what is accomplished by said burning, if anything). The fact that Jesus uses κατακαίω means that He is trying to illustrate a process of burning that is brought to a conclusion with the darnel being fully consumed. This is what the word κατακαίω refers to ... being completely destroyed by fire (see LSJ, Thayer's, and Strong's Concordance).
This is also evident in the Septuagint (LXX). The word is used to refer to sacrifices that have been reduced to ashes (Lev. 4:12, 21; 8:17; Num. 19:5; 17), and to the discarding of the vestiges of unclean parts of sacrifices (Exo. 29:14, 34; Lev. 6:30-31; 7:17; 19; 8:32; 9:11; 16:27; 19:6), and even to the judgement that is administered for specific sins, whereby the transgressors would be executed (Lev. 20:14; 21:9). It is also used when the Israelites are ordered to burn and demolish pagan sites (Deut. 7:5, 25; 12:3), and when the complete destruction of the unrighteous, as well as the desolation of their lands, is in view (Isa. 1:31; 33:11-12; Jer. 21:10; 34:22; Eze. 20:47 cf. Matt. 3:12). Essentially, the word was used in the LXX when something was done away with completely by fire, destroyed, not left intact; not when something was being perpetually burnt and never consumed. Putting all of this together, it is clear that Jesus intended to communicate that the weeds which are gathered are fully burnt up.
II. As the weeds are gathered and burned so it will be at the end of the age
Let's now take a look at Matthew 13:40. Here Jesus clarifies the parable He had told previously. He compares the darnel that is gathered up and burned with fire with what happens at the "end of the age". What happens at the end of the age? Jesus sends out His angels to "separate the wicked from among the righteous" (v.49) by gathering "all things that cause stumbling and those who do iniquity" to toss them into the "furnace of fire", where there will be "weeping and gnashing of teeth". In the parable, the weeds are those who do evil and cause stumbling, hence the gathering of the weeds is analogous to the gathering of the unrighteous, and the burning of the weeds is analogous to them being pitched into the furnace of fire.
And we remember what happens to the weeds, namely, that they are entirely burnt up (note also that the word κατακαίω is again used in v.40. Jesus is clearly putting emphasis on the fact that the weeds are fully consumed by the fire). And so it will be for the wicked who are thrown into the furnace of fire; they will be totally devoured. Apart from the fact that Jesus uses the idea of darnel weeds being incinerated by fire to illustrate what happens to the evildoers who are cast into the fiery furnace, what else can be used to support the notion that the wicked are totally destroyed?
III. Furnace of fire
The expression "furnace of fire" likewise contributes to the idea of the unrighteous being wholly consumed, as Jesus is alluding to the fiery furnace in Daniel 3. Jesus partly quotes Daniel 3:6 from the LXX in Matthew 13:42.
[Matthew 13:42] and they will throw them into the furnace of fire; in that place there will be weeping and gnashing of teeth. (NASB)
[Matthew 13:42] καὶ βαλοῦσιν αὐτοὺς εἰς τὴν κάμινον τοῦ πυρός· ἐκεῖ ἔσται ὁ κλαυθμὸς καὶ ὁ βρυγμὸς τῶν ὀδόντων. (THGNT)
[Daniel 3:6] And anyone who does not fall down and do obeisance, they will throw him into the furnace of fire.” (NETS)
[Daniel 3:6] καὶ ὃς ἂν μὴ πεσὼν προσκυνήσῃ, αὐτῇ τῇ ὥρᾳ ἐμβληθήσεται εἰς τὴν κάμινον τοῦ πυρὸς τὴν καιομένην. (LXX)
Jesus' listeners would have most likely recalled the furnace of fire (κάμινον τοῦ πυρός) from Daniel 3 when Jesus spoke about the unrighteous being tossed into a "furnace of fire" at the end of the age. And if so, what would they have pictured happening to those condemned? Undoubtedly that the fire of the furnace would kill them. After all, that's what happened to the men who had been ordered to throw Shadrach, Meshach, and Abednego into the furnace.
[Daniel 3:22] Because the king’s order was urgent and the furnace overheated, the flame of the fire killed those men who took up Shadrach, Meshach, and Abednego.
Since the furnace was superheated, it incinerated the men from Nebuchadnezzar's army who had taken up Shadrach, Meshach, and Abednego. The only reason this didn't happen to the boys' was that they were protected by God. They were completely unscathed, as the blaze had no effect on them, and they were so conserved because they were righteous servants of God (He shelters the righteous from death and destruction, not the unrighteous; preservation is a sign of God's favour).
[Daniel 3:27] And the satraps, the prefects, the governors, and the king’s counselors gathered together and saw that the fire had not had any power over the bodies of those men. The hair of their heads was not singed, their cloaks were not harmed, and no smell of fire had come upon them.
Jesus could not have meant for His audience to conclude that this would be the same outcome for those thrown into the furnace of fire at the final judgement, since they will not be righteous servants of God, but corrupt and iniquitous. Also, if they are to be untouched by the flames of the furnace, as Shadrach, Meshach, and Abednego were, casting them into it would be futile from a punitive standpoint (the immunity of young men to the intense heat of the furnace was a salvific act on God's part that demonstrated not only their devotion to and trust in Him but also the supremacy and capacity for deliverance possessed by God alone [v.28-29]). Thus, Jesus' disciples would have understood Him to be expressing the demise of those who will be tossed into the furnace of fire at the end of the age, since that is what happened to the soldiers who were not under God's protection when they neared the blazing furnace of Daniel 3, to which Jesus was alluding.
IV. Weeping and gnashing of teeth
And now we will consider the final phrase, "weeping and gnashing of teeth". Many think that it supports ECT. How so? Essentially, it is supposed that the weeping and gnashing of teeth go on for eternity and then concluded that the wicked must remain in anguish for eternity. The latter would follow if the former was true, but nowhere in scripture suggests that it is. If you search for all occurrences of "weeping and gnashing of teeth" (Matt. 8:12; 13:42; 50; 22:13; 24:51; 25:30; Luke 13:28), you will see that none of them says that it goes on forever. It is an unwarranted presupposition, erroneously imposed upon the text by those reading it through a traditionalist lens. It never mentions how long it lasts, just that it happens.
But what does "weeping and gnashing of teeth" indicate? Probably not physical suffering, but rather great sorrow and outrage. Gnashing of teeth is an idiom that conveys anger and bitterness, not pain (Job 16:9, Psa. 35:16; 37:12; 112:10; Acts 7:54). And weeping can imply either physical pain or grief and anguish. I think it is the latter since the weeping and gnashing of teeth does not only occur when the unrepentant are hurled into the furnace of fire but also when the "sons of the kingdom" are cast into the "outer darkness" (Matt. 8:12). But either way, unless God intends on merely snuffing the wicked out of existence instantaneously, they will die violently and painfully and will have more than enough time to express emotions of pain, regret, sorrow, bitterness, outrage, etc. They will surely feel tremendous grief, anguish, wrath, and even resentment when they become aware of their impending doom and irrevocable destruction, and how, unlike the saved, they will permanently lose out on an immaculate and unending life free of all suffering and evil in a perfectly restored cosmos together with their Creator, their Lord, and all of sinless humanity.