While we can't identify the fungus to genus or species level (there are hundreds of wood decay fungi, and some can look very similar to each other without microscopic examination), you probably won't be exposed to anything hazardous if it's burned. That said, it might cause lots of fungal spores to become airborne that may bother anyone with a mold or similar allergy/sensitivity. While spores drifting around the house won't necessarily cause any problems for wood surfaces or structural wood (because it's too dry for the fungus to invade), you still might want to try cleaving off the fungal-ridden portions of such logs if you want to burn them. Logs with fungal growth have already begun to decay, so they might not burn as long or as hot, and fungi need a certain level of moisture in the wood to grow within it, so the presence of mushrooms also suggests that the wood is not as dry as is ideal for burning or storage without further degradation.

Miri