Is it safe to burn mouldy wood?

Wood should be “dry” when you burn it. I put dry in quotes because unless it is stored in outer space it is never totally dry. In winter it might be 4% water, in summer it might be 12% . You can easily watch a logs mass go up and down by weighing it on a good scale.
With that as background…
Yes. As long as the wood is reasonably dry it does not matter how it looks. If it is dirty or dusty slip it into a paper bag and burn it in your wood burning stove. Don’t waste the BTUs!

To follow up on @LuckyGuy‘s generally good advice, if the wood has “pockets” of mold or dampness, then these could cause a generally unfavorable condition as the water trapped inside the solid wood expands rapidly to steam – in a hot stove or fireplace – and could “explode” the log. That is all dependent upon how much water there is, whether it has an avenue to “steam off” safely as it heats up, the temperature of the stove or fireplace, etc.

And the danger, if there is any, can also be mitigated by the stove design and whether all openings are properly covered so that the pressure from any rapid expansion will be vented through the flue, as it should be, and not onto the hearth or floor of the room.

That’s all a judgment call. The general advice holds: burn dry wood.

Another area of concern – the makeup of the fungus or mold that is growing on the wood. If the wood is just wet, that’s one thing. If there is fungus mold (usually, but not always a darkish green) then you have potential danger from the mold itself.

Yes, the fire may kill the mold, but it may also heat it and cause spores to be released into the stove (if enclosed) or into the room (if burning in a fireplace with only a screen). So there’s more to worry about than just whether it will burn.

I have a high efficiency Lopi Freedom stove with integral reburner. It runs considerably hotter and cleaner and more efficient than “normal” stoves with a single baffle.
I burn anything that is reasonable dry. I don’t care what it looks like or it’s composition. The stove eats it and burns it cleanly.

@LuckyGuy – that’s a stove. If @longgone is burning in a fireplace, it’s may not be enclosed and it almost definitely won’t be as hot.

@longgone – what can you tell us about how you burn?

Thanks, Fluther friends. I plan to burn my wood in an fire bowl, outside, cooking food over it.