How can I cure exfoliative cheilitis (dead skin shedding from lips)?

It does depend on what, exactly, has caused it and what exacerbates the symptoms.

It is possible that an anti-fungal cream might help, if the problem originated through a yeast infection. If it seems to be stress-related, that should be addressed, as a topical cream would only sooth the surface of the lips. It could also be caused by low Vitamin B12 and folate, zinc or iron, but you would probably have noticed other symptoms; poor circulation, tiredness, irritability, hair loss. There are auto-immune disorders that might prevent healing in general, so that is also possible. And hormone imbalances can also present with this problem.

You may have already checked all the above and still have symptoms, in which case, make sure you cover your lips with a good quality lip balm during UV exposure, whatever the original cause. Try not to touch the area, or lick your lips as this will dry them more and increase the risks of inflammation and introducing bacteria into the area.

You could try extract of calendula, it is a very good skin healing extract and has had some impressive results in a 10% solution. Chamomile is very soothing ( it contains various bioactive phytochemicals - but don't use it if you are allergic to chrysanthemum), and both yarrow or evening primrose could help too - as extracts or dab on a hydrosol. A physician may suggest a hydrocortisone cream.

Oils probably won’t help much in the long-run and applying a wax-based lip-balm won’t sort the underlying cause and you will only have relief from symptoms for a short time after applying it. So, if you’ve already eliminated the more common causes, I would try calendula.