How can I insulate the inside of a fire escape door?
I think you only possible solution, that would not violate life/safety codes, would be to build a portico over the door outside to keep the sun off of it for as long as possible. 



The other option is to paint the door exterior with a special heat reflective paint. It is designed to reflect a lot of the infrared energy away from the surface it is painted onto. I have put it on steel shipping containers used as site offices, it is highly effective. It has no effect if painted on the inside, it has to go on the exterior. So both options will likely require some interaction with your landlord.