An eczematous cheilitis possibly relating to atopic tendency . A patch testing to identify and avoid potential allergen is indicated.
Firstly, It can be due to vit B deficiency. Take some vitamin B complex for some days and see if your symptoms improve.
Secondly it can be due to cheilitis. Thirdly, it can be a contact dermatitis. Take some over the counter antihistaminic like benadryl or Claritin and see if it helps.
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.You can also use a homemade ointment mixed from 1 tsp glycerine, 1 tsp lemon juice and 1 tsp castor oil. Spread on lips overnight, refrigerating the leftovers for future use. Eat foods high in calcium, magnesium and flaxseed oil. 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.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.