Hello! The condition you described is known as lip smacking dermatitis. It commonly happens when the lips are dry.
You can use a mometasone cream. Apply it twice a day for three days and also intermittently use a lip balm.
Next Steps
Use lip balm for at least a week
Health Tips
Avoid lip balms which have lot of fragrance as it causes sensitisation and lead to burning. Ghee for that matter also works very well.