My DS has this happen for the first few weeks of life. I don't know if it was due to latching issues (as far as I know our latch was fine) or what...but it did resolve itself on its own. I put Aquaphor on it when it looked especially bad. But he never had any pain from it and it just stopped happening after about 3 weeks (I think...he's going to be 7 weeks this week and it's been several since I noticed it)
This happened to both of my LOs, it usually started after a long, wet nursing session. It resolved after a week or two (happened a couple of time during that period). But my LOs peeled all over their body- lips included. The new skin was beautiful underneath!
My DS is 10 months old and we still have this problem periodically. I usually just let it be or put some coconut oil on his lips if it looks really bad. It was worse in the winter time or when he was nursing more than usual. Definitely nothing to worry about.