Repairing a hole in a door
You can try something similar to the fix for a drywall hole, just with glue instead of screws due to how thin the material is:


Cleanup and smooth the opening and make a replacement piece that fits the hole.
Install a small piece of wood across the back side of the opening.  You need this to attach your patch to.  Use a strong glue (liquid nail perhaps) to have it stick in place.  Because of how thin the material is, you may want to install a few pieces to provide support at the edges.  If you don't have a small piece of wood, pickup a paint stirrer at the hardware store, they're usually free.
Glue the patch in place.
Wood filler or spackle around the edges.
Sand.
Prime and paint.