How do I fill in holes in a fire door?
If you could dimple in the metal around the hole so that you leave enough metal for screw threads to engage then you can install a flat head sheet metal screw that has threads right up to the head. This strategy in the best case would recess the screw head enough so that you could use some high temperature epoxy to fill in over the screw head. In the less than optimum case the screw head may stick out above the surface in which case you would have to go ahead and grind it down to flush or more so that you can the fill the area around and over the screw head. 

If the metal door has a wood core behind where holes are located you may be able to countersink and dimple the area around the hole so that a flat head screw will fit up to the surface. In this case a longer screw could be threaded into the wooden core inside the metal door.

If there is no real chance to get a screw threads to engage in the metal cladding of the door or into some door core material then the problem of plugging the hole with a metal piece becomes a bit more complicated. For a very small hole in the size of say 1/8 inch it would seem to me that a high temperature epoxy may be the best thing. Just fill it, let it harden and then sand to level. Note that when using the filler it would really be best to sand down to bare metal near the hole. The epoxy will adhere better to bare roughened bare metal than it would to paint surface.