I cannot confirm anything without examination but it can be contact dermatitis, cheilitis or herpetic gingivostaomatitis(caused by herpes virus).
Take some vitamin B complex for some days and see if your symptoms improve. Avoid licking the lips as saliva evaporates quickly resulting in them being drier than before. Drink plenty of water and take multivitamin supplements and do not apply lipstick or other cosmetic products for some days. Use a humidifier to moisturize the air in your home and apply shea butter or Vaseline on your lips. If a week of compresses and emollients don't do the trick, you can add over-the-counter 1 percent hydrocortisone ointment to your regimen.
If the condition still persists or there is deep cracking in the region of your mouth, then please consult your doctor as it can be due to infection(Angular Cheilitis).Antibiotics are needed in such cases. I hope it helps. Take care and God bless.