Short answer: The burning coals are the vengeance that God has promised to hurl down upon Jerusalem for all of the sins of the Jews since Sinai. By "turning the other cheek" the believer "leaves room" for divine wrath:
[Jude 1:7-9 ESV] (7) just as Sodom and Gomorrah and the surrounding cities, which likewise indulged in sexual immorality and pursued unnatural desire, serve as an example by undergoing a punishment of eternal fire. (8) Yet in like manner these people also, relying on their dreams, defile the flesh, reject authority, and blaspheme the glorious ones. (9) But when the archangel Michael, contending with the devil, was disputing about the body of Moses, he did not presume to pronounce a blasphemous judgment, but said, "The Lord rebuke you."
In Rome, the Jews and gentiles who believe are being persecuted by the unbelieving Jews. The new covenant saints are being persecuted by the old covenant saints:
[Gal 4:21-31 ESV] (21) Tell me, you who desire to be under the law, do you not listen to the law? (22) For it is written that Abraham had two sons, one by a slave woman and one by a free woman. (23) But the son of the slave was born according to the flesh, while the son of the free woman was born through promise. (24) Now this may be interpreted allegorically: these women are two covenants. One is from Mount Sinai, bearing children for slavery; she is Hagar. (25) Now Hagar is Mount Sinai in Arabia; she corresponds to the present Jerusalem, for she is in slavery with her children. (26) But the Jerusalem above is free, and she is our mother. (27) For it is written, "Rejoice, O barren one who does not bear; break forth and cry aloud, you who are not in labor! For the children of the desolate one will be more than those of the one who has a husband." (28) Now you, brothers, like Isaac, are children of promise. (29) But just as at that time he who was born according to the flesh persecuted him who was born according to the Spirit, so also it is now. (30) But what does the Scripture say? "Cast out the slave woman and her son, for the son of the slave woman shall not inherit with the son of the free woman." (31) So, brothers, we are not children of the slave but of the free woman.
God is in many ways a lot like Google:
- he searches the heart, the kidneys, the lungs, your sitting down and standing up, your eating and drinking, and all things
- he stores up everything you say or do on his servers in the "cloud" including your texts, your tears, your emails, the schemes you make as you lie in bed, your words, your deeds
- he appoints a day of visitation
God told Israel long ago that he was going to store up the judgment for their sins:
[Deu 32:34-35 NLT] (34) "The LORD says, 'Am I not storing up these things, sealing them away in my treasury? (35) I will take revenge; I will pay them back. In due time their feet will slip. Their day of disaster will arrive, and their destiny will overtake them.'
[Deu 32:34-35 LXX] (34) οὐκ ἰδοὺ ταῦτα συνῆκται παρ᾽ ἐμοὶ καὶ ἐσφράγισται ἐν τοῗς θησαυροῗς μου (35) ἐν ἡμέρᾳ ἐκδικήσεως ἀνταποδώσω ἐν καιρῷ ὅταν σφαλῇ ὁ ποὺς αὐτῶν ὅτι ἐγγὺς ἡμέρα ἀπωλείας αὐτῶν καὶ πάρεστιν ἕτοιμα ὑμῗν
So way back in Deuteronomy 32 God announced a day of vengeance upon which he would pour out wrath. John the Baptist, Jesus and the apostles all announced that that day was imminent - at the doors:
[Luk 3:7-9 KJV] (7) Then said he to the multitude that came forth to be baptized of him, O generation of vipers, who hath warned you to flee from the wrath to come? (8) Bring forth therefore fruits worthy of repentance, and begin not to say within yourselves, We have Abraham to our father: for I say unto you, That God is able of these stones to raise up children unto Abraham. (9) And now also the axe is laid unto the root of the trees: every tree therefore which bringeth not forth good fruit is hewn down, and cast into the fire.
[Luk 21:22 NKJV] (22) "For these are the days of vengeance, that all things which are written may be fulfilled.
Jesus said that God was going to send the Jews prophets specifically for the purpose of their being rejected in order to and finally put an end to the offensive earthly city:
[Mat 23:31-38 ESV] (31) Thus you witness against yourselves that you are sons of those who murdered the prophets. (32) Fill up, then, the measure of your fathers. (33) You serpents, you brood of vipers, how are you to escape being sentenced to hell? (34) Therefore I send you prophets and wise men and scribes, some of whom you will kill and crucify, and some you will flog in your synagogues and persecute from town to town, (35) so that on you may come all the righteous blood shed on earth, from the blood of righteous Abel to the blood of Zechariah the son of Barachiah, whom you murdered between the sanctuary and the altar. (36) Truly, I say to you, all these things will come upon this generation. (37) "O Jerusalem, Jerusalem, the city that kills the prophets and stones those who are sent to it! How often would I have gathered your children together as a hen gathers her brood under her wings, and you were not willing! (38) See, your house is left to you desolate.
So Jesus is consigning natural old covenant Jerusalem to a fate in the pattern of Sodom:
[Jude 1:7 ESV] (7) just as Sodom and Gomorrah and the surrounding cities, which likewise indulged in sexual immorality and pursued unnatural desire, serve as an example by undergoing a punishment of eternal fire.
However, this time the fire will come through Rome as they burn the sinful city.
Hence, "burning coals".
In support of this non-redemptive view I cite Paul's own words:
[Rom 2:4-5 NLT] (4) Don't you see how wonderfully kind, tolerant, and patient God is with you? Does this mean nothing to you? Can't you see that his kindness is intended to turn you from your sin? (5) But because you are stubborn and refuse to turn from your sin, you are storing up terrible punishment for yourself. For a day of anger is coming, when God's righteous judgment will be revealed.
So Paul is saying that by "turning the other cheek" the one who persecutes you is adding to the reasons and to the severity of the wrath that will fall upon your persecutors. In this way one "heaps up coals upon their head". IE: coals on your head is not a good thing but a bad thing:
[Psa 140:10 ESV] (10) Let burning coals fall upon them! Let them be cast into fire, into miry pits, no more to rise!
[Pro 25:22 ESV] (22) for you will heap burning coals on his head, and [IE: "but"] the LORD will reward you.
The idea is that rather than respond to evil with evil, respond with goodness so that your enemy will be punished severely for mistreating you while you will be rewarded by God:
[Rom 12:17-21 NET] (17) Do not repay anyone evil for evil; consider what is good before all people. (18) If possible, so far as it depends on you, live peaceably with all people. (19) Do not avenge yourselves, dear friends, but give place to God's wrath, for it is written, "Vengeance is mine, I will repay," says the Lord. (20) Rather, if your enemy is hungry, feed him; if he is thirsty, give him a drink; for in doing this you will be heaping burning coals on his head. (21) Do not be overcome by evil, but overcome evil with good.
[Rom 12:17-21 MGNT] (17) μηδενὶ κακὸν ἀντὶ κακοῦ ἀποδιδόντες προνοούμενοι καλὰ ἐνώπιον πάντων ἀνθρώπων (18) εἰ δυνατόν τὸ ἐξ ὑμῶν μετὰ πάντων ἀνθρώπων εἰρηνεύοντες (19) μὴ ἑαυτοὺς ἐκδικοῦντες ἀγαπητοί ἀλλὰ δότε τόπον τῇ ὀργῇ γέγραπται γάρ ἐμοὶ ἐκδίκησις ἐγὼ ἀνταποδώσω λέγει κύριος (20) ἀλλὰ ἐὰν πεινᾷ ὁ ἐχθρός σου ψώμιζε αὐτόν ἐὰν διψᾷ πότιζε αὐτόν τοῦτο γὰρ ποιῶν ἄνθρακας πυρὸς σωρεύσεις ἐπὶ τὴν κεφαλὴν αὐτοῦ (21) μὴ νικῶ ὑπὸ τοῦ κακοῦ ἀλλὰ νίκα ἐν τῷ ἀγαθῷ τὸ κακόν
However, Peter does see a possibility for redemption coming out of it when the sinner beholds, as it were, in the saints, the suffering of Christ:
[1Pe 2:18-25 NLT] (18) You who are slaves must accept the authority of your masters with all respect. Do what they tell you--not only if they are kind and reasonable, but even if they are cruel. (19) For God is pleased with you when you do what you know is right and patiently endure unfair treatment. (20) Of course, you get no credit for being patient if you are beaten for doing wrong. But if you suffer for doing good and endure it patiently, God is pleased with you. (21) For God called you to do good, even if it means suffering, just as Christ suffered for you. He is your example, and you must follow in his steps. (22) He never sinned, nor ever deceived anyone. (23) He did not retaliate when he was insulted, nor threaten revenge when he suffered. He left his case in the hands of God, who always judges fairly. (24) He personally carried our sins in his body on the cross so that we can be dead to sin and live for what is right. By his wounds you are healed. (25) Once you were like sheep who wandered away. But now you have turned to your Shepherd, the Guardian of your souls.
[2Ti 2:24-26 NKJV] (24) And a servant of the Lord must not quarrel but be gentle to all, able to teach, patient, (25) **in humility correcting those who are in opposition, if God perhaps will grant them repentance, so that they may know the truth, (26) and that they may come to their senses and escape the snare of the devil, having been taken captive by him to do his will.
If you want to get back at your enemies then do them good. You will suffer innocently and God will reward you and vindicate you, pouring wrath on your enemies.