How long do burnt trees stay around?

Burnt trees, like the stumps of chopped ones, are out of the normal tree lifecycle. This means that a burnt tree will remain in that state indefinitely, until chopped.

Once a tree gets burned down it basically becomes a new gatherable resource (charcoal) in the same way you gather twigs from saplings, with the difference that you can't plant a burnt tree.

Anyway, if you have some space left in your inventory and you've lost many trees in the fire, it's advisable to gather the charcoal instead of waiting, since chopping down those trees has a (small) chance to yield some pine cones to help you regrow a forest in a short time.