You'd normally use "coax" when encouraging a scared animal to come to you, usually attracting it with treats. When you coax someone/something, You're persuading them/it on a very rudimentary level, but also very nicely. "Coax a fire" is figurative. It tells me the flame is shy and reluctant to begin doing what you want it to, like a frightened animal. Offer tinder as a treat. ^_^

"Coax" means to encourage or persuade. To "coax a fire" would be to start a fire or to try to start a fire (by means of blowing on a small flame that has been lit around twigs and paper, for instance).