What is causing the fat pockets above my eyes? (photos)

You have a very shallow orbit, The place where the eyeball sits.  The "fat" that you're referring to is most likely your lacrimal gland. You need to have an evaluation   For thyroid disease. If your eye continues to prolapse, this could cause damage to your optic nerve. There are several ways to remedy this problem. You need to see an orbital specialist. 

Without examing you, it would appear from your photos that you have ptotic or prolapsed lacrimal(tear ) glands rather than fat deposits. Your symtoms of eye irritation,burning, etc, are consistent with long standing dry eye syndrome. I would recommend consultation with a good opthalmologist to fully evaluate your eyes, and would not recommend any cosmetic procedures on them until you have done so.

This needs to be assessed by an ophthalmologist or oculoplastic surgeon.  Your photos suggest a droopy or enlarged lacrimal gland.  There are systemic medical problems that can present this way.  For that reason, you should pay attention to these signs and symptoms and get yourself assessed.