I've had chapped lips for about a year now. I have eczema on my arms and I think it may somehow be connected. I've visited a dermatologist but none of his stuff has worked. The only thing that works is chapstick but that only lasts for about 2 hours. It's usually just chapped for about half of the day, and sometimes it will be fine for almost a week.
Your name does not let me know whether you are male or female. If you are female it could be dye in lipstick. It happened to me many years ago. If this is the case, do not use lipstick for a long time and let it clear. Try Revelon lipstick, so far I am getting away with this brand.