You have water flow through the heater core!
The heater valve (monovalve) is powered when the key is on. It is held closed when the pushbutton controller grounds the other side. Other than internal problems with the valve, I would look to see if the whole circuit is complete. Verify battery voltage on one side and ground on the other. For test purposes ground the control side or pinch off the water supply with some shielded visegrips (take a piece of old hose and slit it, put around water hose and squeeze on this to protect the water hose).
Put a cork in it.
Bosch Master, ASE Master, L1
33 years MB technician