Wood burner is the skill associated with the wood burning labor, where dwarves turn wood into either ash or charcoal at a wood furnace. Dwarves with high skill level accomplish these tasks more quickly. Typically, a wood stockpile and a bar/block stockpile are placed near the furnaces.
How do I get dwarves to burn wood?
Wood burner is, somewhat tenuously, considered a farming-related job (because ash can be made into potash, which fertilizes fields) and thus the labor is filed under Farming/Related labor menu, and not furnace operating.