In the first sentence you should avoid repeating the verb remain. Try using stays or continues to stay.
In the second sentence it sounds like the room is the thing remaining unopened. I think you mean door. "2) There is a need to get a door closer installed in our room, where we sit. Without the door closer, our room remains opened causing our room (to) remain warm / our room (to be) warmed."
I hope this helps you. I put my corrections in parenthesis.
Interested in Language