My comment:
The matrix is leaking. Air gets sucked into the heater matrix as it cools. When engine is started the water pump pressure forces the air out from the matrix. Have you checked the radiator coolant level - is the coolant well below the radiator cap? Do no rely on the level of the reservior since air in the radiator stops the reservoir topping up the radiator : remove the rad cap and check the coolant level.
Check the carpet to each side of the heater ... my bet is it will be wet indicating the heater matrix is leaking.