Do I need surgery to get rid of a red, peeling, swollen lump left after an eye infection? (Photo)

You should see an ophthalmologist or oculoplastic specialist for evaluation. You may have chronic chalazion that needs to be removed. 

As the initial inflammation settles down, it organizes into a chalazia.  This is a reaction to leakage of oils from an inflamed meibomian gland in the eyelid.  Surgery is only necessary if the lesion fails to respond to more conservative treatment.  The most conservative treatment is what you have been doing.  Now it is reasonable to see an oculoplastic surgeon and have them inject it with kenalog.  This will often complete resolve what is left of the lesion.