Why does hot water come from the faucet when only the cold water tap is on?
There are a few possible reasons:


Some hot water recirculation systems use the cold water as a return. In these systems, warm water is returned in the cold water pipe until hot water reaches the the temperature valve that controls the recirculation. Or, perhaps the one-way valve near the pump is failing.
Hot water pipes could be touching a cold water pipe (probably near the water heater). The two pipes touching would heat up the cold water pipe.
Cold water pipes could be touching a ventilation duct. In the winter, the duct would be warm and would heat up the cold water.
No heat trap in the water heater inlet. Hot water might flow up out of the water heater into the cold water pipes.


In the morning, try to feel which parts of the cold water pipe are hot, and that'll be your answer.

Another thing to check, but is likely unrelated, is the condition of a water expansion tank (generally mounted near the water heater) if you have one, or contemplate installing one if you do not (they're needed if you have a pressure reducing valve or a one-way valve in your water meter).