Thanks for the insight guys. I repaired the dent and applied about 3/4
inch carpet padding on the interior of the door. There's enough
clearance for it. And it actually does make a noticeable difference in
noise reduction.

As far as enough air for combustion, I don't know how the door could
allow air in except around the seams between the frame and the door. So
I didn't put any padding around the outer edge of the door. It's not a
louvered door but a solid, hollow core. Think I'm safe?