I would not classify this as exfoliative cheilitis. But I am not saying you don't belong here by any means.
This could very well be an allergy. I suggest you get patch testing done, and in the meantime stop using any products that could be causing this (soaps, toothpaste, shampoo etc). Change all of those products to Sodium Lauryl Sulfate Free (SLS Free). That is a good place to start.