Burning of lip and face can be due to dryness, allergy, vitamin B 12 deficiency or due to peripheral neuropathy due to diabetes, hypothyroidism, Lyme’s or lupus.
Dehydration is the commonest cause of dry cracked lips. Drink plenty of water and see. The other possibility is iron and Vit B deficiency. A lip balm preferably petroleum jelly, Vit B supplements, iron, and honey should help. They can also be caused by allergies and infections. You could either be suffering from contact dermatitis or from allergy to some food. You should get an allergy test done, either a skin test or RAST, to know what you are allergic to. Contact dermatitis could be due to tooth pastes, mouth cleansers, spoons, cups, glasses, food items, cosmetics, shampoos, etc, basically anything that comes in contact with your lips.
Another possibility is eczema. Herpes infection is another possibility if there is a history of kissing or sexual contact (can get aggravated in presence of vitamin deficiencies, dry cracked lips, infections etc). Also get your blood sugar checked both fasting and two hour after food to know what if you have diabetes. Hypothyroidism too can cause this type of skin and lip problem. I would suggest you consult a skin specialist regarding this. Take care!