Sounds like a conk. They will be fine in a vivarium and will last just like wood if they are fully dry. If they get soggy they can mold. And they can possibly drop some spores which might become new baby conks on wood decorations. These living conks are perfectly fine, and you can think of them as cool frog perches. Since they are alive they will not rot or breakdown and are preferable to dead "decoration" conks.
Now if they are stemmed mushrooms, even big woody ones, I would pass on putting them into a humid environment.