Crazy news! In a book called Burn, burns end up being super-symbolically important!
Sarcasm aside: true facts. Burn is burning up with burns (and other scars). At the character level, burns/scar are symbolic of how characters relate to the past. Pressia wants to have her scars removed; she believes that the past can be mended. Bradwell doesn't want his scars removed (or his awesome wings); he believes, like Faulkner, that "the past isn't dead. It isn't even past.
And we're not just referring to the scars and burns of the past: these characters, what with all their airship-crashing, Dome-invading and Dust-repelling, just keep accumulating. Unsurprisingly, the character that seems the most gung-ho about collecting all the scars possible is El Capitan:
"Scars are good. Right, Helmud? It's the body's way of making armor." (3.12)
El Capitan: a firm believer in the motto "what doesn't kill you makes you stronger."
Basically, if you want a shortcut to deep insight into these characters, look at the way they handle their scars and burns… because scars and burns in Burn symbolize not only the past, but the hardship of the present.