The Sages enacted a prohibition against medical procedures on Shabbat, due to the concern that people might grind herbs to produce medicine, which would constitute an act of Shabbat desecration. People in ancient times would commonly grind various herbs in order to produce medicine, which is why, incidentally, the common pharmaceutical symbol to this very day is a mortar and pestle. The Sages therefore forbade taking medication on Shabbat, except under certain conditions, to avoid the risk of people desecrating Shabbat by producing medicine.
In light of this prohibition, it is forbidden to apply oil to chapped skin – such as on the hands or lips – on Shabbat. Applying creams and lotions is forbidden even irrespective of this prohibition, due to the separate prohibition of Memare’ah (smoothing out a thick substance). But even liquids, such as oils, may not be applied to skin if this is done for medicinal purposes. This is the ruling of Hacham Bension Abba Shaul (Israel, 1924-1998), in his work Or Le’sion (vol. 2, 35:7), and of Rabbi Moshe Halevi (Israel, 1961-2000), in his Menuhat Ahaba.
Hacham Ovadia Yosef makes an exception in the case of infants, noting that oil is often applied to babies’ skin for the sake of cleanliness when they are changed. But for everybody else, since people do not normally apply oil for purposes other than healing chapped skin, this would be forbidden on Shabbat.
Summary: It is forbidden to apply oil to chapped skin – such as chapped hands or lips – on Shabbat. It is permissible, however, to apply oil to an infant’s skin for the purpose of cleaning the skin.