In Romans chapter 12 Paul is discussing how the believer is meant to interact with other people. Specifically, with fellow believers in verses 1-13, and with non-believers in 14-21. When the reader arrives at verse 20, Paul uses a phrase that might catch the reader off guard. He says that for those who treat the believer wrongly, that same believer is treat him with grace and kindness, to the point of providing for the agitators physical needs such as food and water. Then Paul says that by doing so, the believer will “heap burning coals on [the nonbeliever’s] head.”
Given the context of the believer being required to go above and beyond in showing kindness to those who persecute him, the language of this statement stands out by invoking imagery of fire and torment. So what does Paul mean by it? For the longest time I thought that maybe the idea behind the statement is that when you are persecuted, or are being treated wrongly, treating that individual with kindness would induce a sense of spite in them. Their attempts to attack you being met with kindness and compassion would “burn them up,” so to speak. For the longest time, this was the only way I could make sense of the statement. But at the same time, it just didn’t seem right, it didn’t seem very “Christian.” But this week I was doing some studying and it was finally made clear to me.
Jerry Hullinger gives this insight on the passage: “Paul states in 12.20 that when we are kind to others we “heap burning coals on his head.” This phrase occurs in a larger context which deals with doing good things for one’s enemies and leaving their judgment to God. The imagery of “coals of fire” is to be related to the Old Testament symbolism of divine anger and judgment… The idea of the passage then is that Christians can show kindness to the unbeliever and take comfort in the fact that if this kindness is spurned, God will render future judgment.” 
The point that needs to be driven home, I feel, is that Paul gives no opportunity for the believer to take any action that is not done in pursuit of peace and goodwill with those around him. This is after all, the repeated emphasis of verses 14-21. “Live in harmony with one another” (16). “Repay no one evil for evil” (17) “Live peaceably with all” (18).
Paul makes it clear to us that we should not allow ourselves to succumb to feelings of wrath or ill will towards those who wrong us. Rather, what he tells us is that these sentiments should be thought of as being reserved for God: “Never avenge yourselves, but leave it to the wrath of God, for it is written, ‘Vengeance is mine, I will repay, says the Lord'” (19). When an individual sins, whether against a fellow human or not, ultimately the one being sinned against is God. And He alone reserves the right to execute justice. The believer is commanded to pursue peace and harmony.
 Hullinger, Jerry M. “New Testament Culture and Belief: A Study of History, Culture, and Meaning.” 366.