Hole in subfloor under stove: Do I fix? If so do I put flooring over it?
I would replace the subfloor by floating some scrap lumber under two edges of the opening and screwing them in place. Then screw down the patch. 

I'd then run tile down both sides of the stove opening and halfway back from the front, but leave the patched area uncovered in case you need later access. 

This gives you the impression of a completed floor from the front and through any gaps along the sides of the stove.