Well sometimes burning and dry lips are caused due to sun exposure, and sometimes from an unconscious lip-licking habit,breathing with an open mouth,less intake of water.
Apart from this other possibility include Allergic Dermatitis
of your lips.
This may be secondary to generalised systemic disease or may be secondary to local cosmetics or applications you are using.
Stop all Vaseline /lip balm products.
It is advised to apply unflavored lip balm, avoid flavored toothpaste, aloe vera gel, rub cucumber slices on the lips.
Drink plenty of fluids.
Avoid spicy and hot foods.
Avoid exposure to wind, sun, or cold, dry air,breathing with an open mouth and licking your lips.
You can take OTC Cetrizine .
You shoud apply a mild topical hydrocortisone cream to help decrease the irritation.
Avoid lipstick for sometime.
Honey has antibacterial properties which can help heal and soothe chapped lips
If the symptoms 'll persist you should consult a doctor/dermatologist for further evaluation and management.
I hope this helps.
If you have any further query then feel free to ask.