Only use local firewood. Make sure that you use wood from the local area, if you bring it with you from somewhere else you could unknowingly be bringing diseases or insects that kill trees into the area. In this context, local means the closest convenient source.
In addition to only using local firewood, how and where you gather the wood, especially out on the trail, is really important. Only use wood that you find on the ground, don’t cut down living or even dead trees that are still standing.