I appreciate the suggestions. I especially appreciate the reminder on the fluid dynamics issue. There will need to be some air going in if I'm going to pump air out.
Maybe I wasn't clear enough on the rest, though.
I'm not going to change anything. I should expand that to mean I don't want to rip apart the door, either. My projects are all budget driven. If I end up spending more for a new door (plus the hassle of running to Home Depot right before I move out of this rental) than the dough I'll save by grabbing this wasted hot air, it's not really cost efficient. I can spend a few bucks on a small fan and a few batteries. That probably won't even save me the cost of the fans and the batteries. But I can live with that.
There's already a vent there. And I'm not really into the idea of taking off the door (or any part of it) because behind the door is my furnace and water heater. The door keeps the noise out of the living room. It also keeps it so we don't have to look at ugly appliances. So as you might guess I'm really not in to the idea of fiddling with that door. Any other setting, I might.
I will be going to radio shack to look for a small fan that can be powered by a battery.
I'm thinking something like this.
1 or 2 should do it. It will be quiet. Now I just need to find out how to power it with a AA or D battery, or whatever works.
Finally, I want it to be battery powered because there is no outlet in the room. And since it's a rental and I am trying to do it as cost-efficiently as possible, I won't be running a power line in to the room for this project.
Once again, thanks for the suggestions.