If you shut the water off coming in, and then let the thing sit, does the water level in the tank go down? If so, there are only a few ways that can happen that I can think of:
- leaking from underneath the flapper
- leaking from underneath the flapper seat
- leaking through a crack in the overflow tube
- leaking through the tank bolts
- leaking through a crack in the tank
The first three, you'd not notice any leaks, as they generally end up just going down the drain.
If the level doesn't go down, but the fill valve runs (and it's not overflowing), it could be siphoning because the refill hose is down the overflow. It should have an air gap above the overflow tube.