The song is a rap battle between Dan Bull and The Stupendium representing different strategies in the game Frostpunk. The goal of Frostpunk is to build a city during a volcanic winter. The Wikipedia page for the game explains:
the player also has the option to increase the citizens' support either by "Order" which includes buildings and laws to enforce security, or by "Faith", which includes buildings and laws that implement a religion. These two paths can be continued to the point of fanaticism, with the "Order" path leading to a militaristic autocracy, while the "Faith" path leads to a total theocracy.
The song's two rappers represent Faith and Order. The speaker of these lines is the chief of the Faith path. Since the entire earth is caught in volcanic winter, everybody is shivering, but the rapper says Each shiver will deliver us. This refers to the first meaning of deliver in Merriam-Webster:
1 : to set free
// and lead us not into temptation, but deliver us from evil
— Matthew 6:13 (King James Version)
The idea is that inescapable physical discomfort indicated by shivering will force the believers to seek spiritual consolation, which will ultimately (in time) be beneficial as it will draw the believers closer to God.
The followers of the Order path believe that anybody who steps out of line should be harshly punished. Earlier in the song, the Order leader says:
To keep the fire alight, we need to stamp all defiance out.
That is, those who follow Faith rather than Order need to be exterminated to keep the fire alight. The implication here is that members of the Order will burn alive members of the Faith, which will not only remove their enemies but also provide light and warmth in the winter.
The Faith leader is saying that the followers of his cult could pre-empt the Order by burn[ing] the innocents for penitence. The innocents are members of the Faith. They can self-immolate as an act of defiance against the Order. This would also be penitence because it would be amends for having sinned against God. The Faith leader has spoken earlier of grace and the ark. God flooded the world for its sinfulness but spared Noah through his ark. The theology here is that humans are inherently sinful and deserve punishment, but are saved only by God's grace. Self-immolation would be a way to repent for the sins against God that have caused the persistent winter.