I have been looking at this forum for a long time and this is my first post. I am going to try to help you.
Firstly I can tell you that a lot of people got this condition because of ACCUTANE. It kills the skin of the lips and the face and when people are on this condition they try to peel off the skin of the lips. This is the last thing to do !
If you peel your lips you MUST stop doing it. My tip is to put a strong lip balm (cortibalm) throughout the day and when I don't need to apply it I leave my lips alone. If you have skin that is ready to come off you can slighly cut it with a nails clipper or a small cisors.
I have done this method for 2 months and they are already so much better even if I am not cure.
If you don't peel your lips, it can be a completely different cause and you have to find it by yourself such as (allergy, thyroid ect...)