I am unable to remove the flush valve. I've had the whole tank off, trying to just hand twist it out. No luck, and I'm afraid to try too hard, thinking the tall thing will break off. The toilet is running, intermittently, as if it were leaking. I can't see any water coming into the bowl, but that's where it has to go... the floor is dry.

You're going about it the wrong way. The way to remove a flush valve is to unthread the large nut on the bottom. If you can't get the nut off take a hacksaw and cut the nut. Good luck tom

If your tank is not over flowing then the flapper is probably not sealing. Removing the valve is likely an unnecessary step at this point

I believe it is the flapper. In this case it doesn't flap, it rides up and down on the central tube. Remove the fill tube, unscrew the retainer nut, remove the sliding tube and get a new seal. HD carries them, $5 or so. Take the old one to make sure it matches. Snaps over the tube. You will probably find a wear ring on the seal.

The large nut on the bottom is what I can't get off... I will have to buy a larger set of Chan-L locs to remove it if there is no other way, and I can't turn the top part either. Thanks very much for your input.

.I will have to buy a larger set of Chan-L locs to remove it if there is no other way,While Channel-Locs are one way there are others.
1- Take a screw driver and hammer and , like a impact driver , place the screwdriver on the nut and tap it out counter clockwise, **OR**
2- Take a hacksaw and cut the nut and peel it off.
Good luck, tom