Heating the water up from cold will cause the system (cold and hot, if there's no check valve on the cold water inlet to the water heater) to pressurize more. My guess is that the toilet valve is leaking under the added pressure, which means the connection is barely adequate under normal pressure. It likely just needs to be tightened.
This would be the flush valve, not the seal between the toilet base and the sewer. The two nuts holding it to the floor have to deal with the latter connection; check that there isn't some other water connection accessible somewhere (or possibly inaccessible somewhere). I guess it's possible that the fresh water connection may be routed within the pedestal, but I don't think that's at all typical.
