Do you snuggle up real close to the door? R U getting burned? Yes.it gets very hot and I don't know of anyone who has insulated a door, but sure, it would work, but why? We need to respect any device that creates heat, but I'm not sure just insulating the door will make a huge difference.
Don't get burned, or if you already have been burned, I'll bet you don't want to repeat the process. Safety first, hard to type with thick bandages on both hands....let alone feed yourself....and other duties!
good luck insulating it, the inside is 12 inches from a pot of burning coal. Maybe pack it with asbestos and never open it? The outside you can insulate because there's a water barrier between the fire and the outside so your insulating 200 degrees not 2000.