On Yom Kippur, one may not apply oil or anything else meant to nourish the skin, to even a small area of the body (SA 614:1). Obviously, any makeup that may not be applied on Shabbat because of issues of dyeing (Tzove’a) or spreading (Memare’aḥ) may not be applied on Yom Kippur either, as everything prohibited on Shabbat is prohibited on Yom Kippur (Peninei Halakha: Shabbat 14:4).
To relieve itch, one may apply oil in liquid form to his skin (Yoma 77b), as long as he does not violate the prohibition of applying medicine; on Yom Kippur, as on Shabbat, it is rabbinically forbidden for one suffering from minor discomfort to use medicine, lest he grind herbal ingredients to prepare it. However, if healthy people occasionally use this oil, it is not considered medicinal, so one may use it to relieve itch. If the itch is painfully irritating, one may apply a factory-produced medicating oil (Peninei Halakha: Shabbat 28:5).
Perfumes or deodorants that make people smell good may not be used. Since they moisten the area to which they are applied enough that touching it would moisten one’s finger (tofe’aḥ al menat le-hatpi’aḥ), using them is considered a form of washing. However, they may be used to remove a bad odor, just as a person may wash to remove grime or get rid of a bad smell. In both cases, the intention is neither for pleasure nor to refresh (section 2 above). Insect repellent may also be used, since it is meant not for pleasure but to repel pests.
Some forbid using perfume and deodorant, but they do not explain whether this is because of the prohibition of washing or the prohibition of anointing (R. Ben-Zion Abba Shaul; R. Seraya Deblitzky). Piskei Teshuvot states that the problem is one of applying ointment (614:1). Shemesh U-magen similarly states that spraying perfume on the hand is considered applying ointment. However, the prohibition of anointing would seem to apply only to something that is meant to nourish the skin, in which case it would pertain neither to perfume nor to deodorant. Therefore, it seems, the relevant concern is only of washing. Interestingly, Ḥida permits kohanim to wash their hands in water to which rose water has been added for fragrance (Ḥayim She’al 1:74). We see that the fact that something makes a person smell good is not intrinsically prohibited. Therefore, if deodorant leaves enough residue that it could wet something else, it may not be used, as it constitutes washing. However, if it is not that wet, and the goal is simply to remove a bad odor, it is permitted. (This would seem to be the opinion of R. Shlomo Zalman Auerbach in Halikhot Shlomo, Bein Ha-metzarim, ch. 14 n. 56, and is quoted in his name by R. Ovadia Yosef in Ḥazon Ovadia: Arba Ta’aniyot, p. 295, and by R. Avigdor Nebenzahl in Yerushalayim Be-mo’adeha: Bein Ha-metzarim, p. 274.) It seems to me that if a woman is worried that not using perfume will make her smell bad and repel her husband, she may use perfume or roll-on deodorant. Spray deodorant, which does not leave enough residue to wet something else, may also be used on Yom Kippur. However, stick deodorant is prohibited, because applying it is considered spreading (Peninei Halakha: Shabbat 14:5 n. 3).