I want to solder my new guitar pickup to my guitar, is it possible to use a wood burner?

It should get hot enough but Irv is right about the tip. The solder will not stick to it, just roll off. Probably will not be able to tin it either even with regular flux. This means heat transfer will not be very good but you should be able to make it work If this is the only soldering you need to do Radio Shack used to have strips of solder that you could wrap around wires and melt with a match.