How should one deal with chapped lips on shabbos? Is it considered refuah?
These do pose problems of refuah on Shabbos, see the Rema, Orach Chaim 327:1, MB 327:4. I do not believe we could consider it common for healthy people to apply olive oil to their skin/lips so doing so would be forbidden. Vaseline and chapstick have the additional problem of ממרח
1)Focus on not licking you lips. We know this but do it out of habit since it gives momentary relief at the expense of exasperating the problem. If you wait it out in most cases you lips will naturally moisten and wont bother you so much as your body naturally moistens the lips 2)Stay hydrated. http://www.mayoclinic.com/health/chapped-lips/an01440
Peeling skin from one's lips is a Torah prohibition (on Shabbos, Biur Halachah 340:2). In cases on distress ask your Rav whether it can be done with a shinui (and what would constitute a shinui/change), although it seems to methis could cause more harm than good.
Chapstick/vaseline or the like are thick enough to present a serious problem of "memareyach" (spreading/resurfacing thick gels). This would be irrespective of their medicinal use. (And this isn't life-threatening enough to warrant any exceptions.)
Using liquid oil there's no problem of memareyach, but there's still potentially the prohibition on medicinal practices (and chapped lips are usually the type of "maychush b'alma" -- "minor affliction" -- that's subject to the prohibition). The way out of this is if the practice is "normal", not "medicinal."
In Talmudic times, it was normal for healthy people to rub olive oil into their skin (see Mishna Shabbos 14:4), so it would be no problem. But the Ramah (OC 328) observes that in times/places when oil is only applied medicinally, it would be a problem. Thus the Shemiras Shabbos KeHilchasa prohibits applying oil on your chapped lips.
I'm still searching if anyone has any other ways out of this, e.g. can we apply the Mishna's workaround of dipping your bread in oil and getting some on your lips that way?
According to many poskim clear liquid lip-glosses are not a problem of Tzoveah on Shabbos since there is no color. It would need to be liquid enough that it is nozel, it flows, and there is debate regarding how viscous it can be and still be considered nozel. This would be a possible solution for women. For men, since lip gloss is not normally used for anything, there would be two problems, one midrabanan: refuah - using the gloss for medicinal purposes, and the other possibly midioraisa, of Lo yilbosh gever simlas ishah, wearing womens' accoutrements. (Of course since it is not for an accoutrement this may not be a problem but then it is obviously for refuah so it would not be able to be used!)
Since there are differing opinions among current poskim regarding:
A. whether clear lip gloss is tzoveah
B. what level of viscousity determines if something may not be spread on the skin on Shabbos
therefore a posek should be consulted.
Regarding a man or boy, a poseik should be consulted as well. I remember when a student of mine had horribly chapped/sore lips he did (with my intervention) get a heter to dab on blistex. But he was in terrible pain, more than just Maihush B'alma.
Sorry I have not brought written sources, but I have discussed the clear lip gloss for make-up question with various poskim and have heard differing opinions on the matter. Igros Moshe talks about viscousity and mimareah and is very mahmir. I will, b'li neder, try to track down the sources pertaining to a clear liquid lip gloss solution for chapped lips on Shabbos over the next week.