Room Absolute Humidity with Different Temperature

No one can help you because your problem statement is unbounded. Given the condition that T2 (and therefore AH2) are at ambient makes it very easy to postulate a "reductio ad absurdum" argument that in the long run T1 and AH1 will go to ambient.

Why? Because you have not bounded the size of the heater, room, door, heat transfer through the walls, or the ambient conditions. I can easily say that it's a 5 watt heater, the room is 100,000m³, the door is the size of one of the walls, the walls are cardboard, and the ambient is the conditions at the South Pole. Or I could take the equally absurd position that it's a 100kW heater, the room is 1m³, the door is the size of a mouse hole, the walls are perfect insulators, and the ambient is the moon.

Be an engineer, or at least a physicist, and put some numbers to your problem, dude.