焼く | To roast food, but sometimes used with other stuff, like cross-burning or flag-burning, or even CD-burning. |
焦がす | To burn to a crisp – to scorch – whether intentionally (flame-thrower!) or unintentionally (bad cook). Also, often used with hot, passionate emotions:burning with the fiiiire of looooooveeeee!!!!!!!!!!!! |
燃える | Incinerate, destroy with fire. You’ll often see signs on garbage cans for 燃えるごみ! |