Oak sat on the ground all summer and fall - now has mshrooms - burn it?

Oak tends to take a while before the inner "meat" becomes punky. I would split some and see how it looks, but I'm guessing it's fine.

I'd dry it and use it
All you are doing is cooking the mushrooms !

So long as the wood was solid when you split it, it's absolutely no issue as long as you get it under cover for a couple weeks. In early November I put about 3/4 cord of oak in my wood shed that was growing shrooms from the wet fall. Two weeks later they were dry and crunchy.

Even a little punk is no big deal, especially in the sap wood. Just get it out of the weather.

Even the dried punky stuff is better than some other firewoods. Definitely keep it

I see a slight drop off curve to BTU output or I should say it burns faster but the other side of the coin is it generally dries quicker. It becomes available for use quicker.
I think its a very fair trade off.

Oak sapwood goes quickly to punk but the heartwood will last a long time. If the rounds are pretty big, that's a good heartwood-to-sapwood ratio, and would be worth grabbing.

When the outside of oak get punky the only draw back is that it gets a little messier when loading in the stove (oak crumbs - if there's such a thing) the inside heart wood usually is as solid as a rock, just make sure its dry so you don't have to fight it.

I've had mushrooms grown on my oak that was stacked on pallets, I just throw them in the stove and burn them.