In taking the cover off my closet dome light to change a lightbulb, I noticed that behind the lightbulb there is peeling tinfoil which is revealing something that looks like insulation. The lightbulbs (2) rest pretty close the tinfoil and the revealed insulation. Is this a fire hazard? Do i need to replace the tinfoil that's peeled away? Is there a max wattage of lightbulb I shouldn't be using?
That is a reflective sheet to help use more of your bulbs' light output. So if you want the same sort of brightness as you had, yep, you do need to replace it. But for a closet, I personally wouldn't bother.
The maximum wattage should be stamped somewhere on the light. If in doubt, use LEDs or similar bulbs that don't push out as much heat as the equivalent CFL or incandescent.