If the hose from the fill valve is not held above the water line by the clip, it can siphon some water out, and that can cause the fill valve to need to run to keep the level up. Verify that the hose is not stuffed down the overflow - it should be clipped above it. If that's not the case, it could be that the flapper valve is leaking, and the fill valve is just keeping up.
Shut off the water supply, then leave it for a few hours. If the water level drops, it's the flapper valve. Sounds like the fill valve needs service or replacement, though, but it may be only one of the problems.