I had the same problem years ago and my dentist told me that a small percentage of people are allergic to tartar control toothpaste and it causes the peeling inside the mouth. Sure enough, it stopped happening when I went back to regular toothpaste. Anytime I have been in a pinch and had to substitute tartar control it happens again. Hope this helps.
i'd see an oral surgeon or periodontist to help diagnose the problem. cold be a myriad of things
Sometimes the tissue can become dry and outer surface will exfoliate. I do not think it is serious but I think it should be seen for a definitive diagnosis. Either your regular dentist or an oral surgeon.