Biotin--- this may be very helpful for those with burning lips.

There are some studies on PubMed that discuss using high dose biotin for burning mouth syndrome. This would be 5 to 10mg doses. Our MS forum is discussing the use of extra high doses of biotin, up to 100mg/day! I am not suggesting anyone here use those super doses, but it illustrates, minimal or no side effects for biotin which is water soluble.

We use 10mg a day for our itchy dry eyes.

Over the past years I've had episodes of upper lip chapping/cracking mostly during winter. I haven't had this lately. We've been using the 10mg biotin for about 2 yrs.
I used a natural type lip balm with the best results. (don't laugh but it is "angry birds" flavored with natural oils. )

Biotin is pretty inexpensive. Puritan's has it up to 10mg and Amazon has Natrol brand in a two pack which we are using now. Biotin is prescribed by dermatologists for hair/nails and skin problems.

I think it is worth trying.

Other causes for burning lips are viral (typically Herpes) and allergic. Some foods cause burning and swelling of the lips, most notably shrimp and other shellfish.
Obviously, hot sauces and some spices may be problematic too for some people. For women, avoiding lipstick may help.
Lipsticks can get contaminated by bacteria quickly. So women may be applying bacteria using partially used lipsticks.

The nerves that go to the mouth and lips are mostly trigeminal ones with some coming up from the lower neck.

If you Google "nerves going to the lips" you can find images on Google Images showing those distributions from both directions.