Don't think I've ever seen any mushrooms growing on it. I have seen some mold/mildew growing on the sapwood only of some staves stored in a damp moldy place.
Just for fun. If you look at Mold under a microscope, many species look like tiny mushrooms. They are related, of course.
Years ago there was an article in the old Primitive Bowhunter Mag, (I think), about curing staves in salt water, too. It seemed to harden or consolidate wood, but then attract moisture later, and everybody talked about that for a few weeks on forums.
I think urine could have a huge anti-fungal benefit. I can't see how water, just water, would actuually improve the wood, unless long times or high mineral content were present. A peat bog would preserve the heck out of a stave,pickling it in tannic acid.
But so far I see the logic behind several points made here. A submerged stave would be far less prone to decay or even oxidative breakdown than one left in the woods. Very wet wood could be much easier to rough shape with stone tools. Checking could be minimized. Nomadic people could store lots of good staves this way and protect them from theft, rot, bugs, animals, etc.. (Imagine finding a PERFECT stave, but already having a really nice bow. You can either remember where the living tree stands, and hope nobody else takes it, nothing happens to it, etc, or cut it down and bury it in the marsh.)
And, so far there doesn't seem to me to be any reasons this is a BAD idea.