We will discuss specifically the Yom Kippur elements of the question; we will not touch upon the question of possible Shabbat violations regarding the use of soap (which is relevant particularly when one wishes to use hard, and perhaps also thick liquid, soap).
It is a matter of debate whether the “afflictions” (i.e., refraining from physical pleasures on Yom Kippur) other than eating and drinking, such as rechitza[1] and sicha,[2] are of Torah origin or are Rabbinic.[3] Either way, built into the prohibition of rechitza is permission to wash when one does so not for enjoyment but to remove dirt.[4] We can safely posit that the hygienic need to wash hands after use of the toilet is no less significant than the need to remove dirt.
Nevertheless, your question is a good one, because if washing hands with soap is considered sicha, it might be prohibited even when simple washing is permitted. This is because we are more stringent regarding sicha than we are regarding washing with water. Indeed, the Yerushalmi[5] teaches that sicha is forbidden even when it is not done for pleasure. On the other hand, the gemara[6] does permit putting oil on chatatim (a type of skin disorder) or for the needs of the sick.
Thus, we must ask: 1) Is using soap classified as sicha? 2) If so, does the need for soap justify its use?
When the gemara[7] discusses sicha, it does so in reference to using oil. Tosafot[8] assumes that the prohibition on Yom Kippur similarly applies to smearing animal fats on one’s skin. The two main ingredients in solid soap are vegetable oils and/or animal fat (although liquid soaps, which we use because of possible Shabbat violations, have more diverse ingredient options). Nevertheless, soaps seem to be fundamentally different from the oils and fats discussed in the classical sources. The objective of sicha is usually to have the skin absorb the substance. Indeed, this is evident from the gemara’s[9] portrayal of sicha as being like drinking. In contrast, soap is intended to be applied and then removed soon thereafter, with only a tiny amount being absorbed by the skin. Thus, one could argue that using soap is not classified as sicha.
However, we find very influential Acharonim – including the Mishna Berura[10] and the Aruch HaShulchan[11] – who assume that using soap is indeed considered sicha. The context of the Aruch HaShulchan’s comment is a discussion of whether it is permitted to use non-kosher soap, which, he writes, depends on how far we take the equation between sicha and drinking. On that topic, Rav Ovadia Yosef[12] notes, in addition to other reasons for leniency, the idea that using soap is “less than regular sicha, as it is immediately washed away with water.” It is unclear whether he intended that such logic alone could make using soap permitted on Yom Kippur, but the statement does corroborate the salience of our distinction and contributes at least slightly to the grounds for leniency.
Why do we treat sicha more stringently than washing, in that the former is forbidden even when there is no intention for enjoyment? A number of explanations are offered. The Magen Avraham[13] writes that it is because sicha generally provides greater enjoyment than washing. Rabbeinu Mano’ach[14] writes that since most people wash off dirt with water, using oil looks like it is being done for enjoyment. Similarly, Tosafot Yom HaKippurim[15] writes that since one could have used water by itself, if he chooses to also use unnecessary oil, it is considered sicha done for pleasure.
Let us now analyze whether these explanations as to why sicha is treated more stringently apply to use of soap in a normal hygienic context. Soap is rarely used for enjoyment. In addition, people are unlikely to be confused about the intentions of one who washes his hands with soap. Finally, soap serves a hygienic function that water does not. Therefore, logic would seem to allow its use for hygiene, just as water is permitted for that purpose. Nevertheless, it is quite possible that anything that falls under the category of sicha is forbidden even if it does not share the classic reasons for stringency. The Mishna Berura[16] might imply this, as does the common ruling/practice forbidding roll-on deodorant on Yom Kippur.
Still, the above argument makes it easier to permit the use of soap based on need. The Shulchan Aruch[17] rules that it is permitted to do sicha for a sick person. Logically, it should be similarly permitted to do so to prevent disease, which health experts assert soap does. However, halachic precedent is potentially influential, and whereas the needs of one who is already sick are well-established in Halacha, the hygienic needs of a healthy person are not.
It is further clear that circumstances impact the degree of the need to use soap, as not all hygienic needs are of the same level. It would also seem to be proper to be consistent on the matter. It does not appear appropriate for one who is not regularly careful about using soap when leaving the bathroom to take advantage of the possibility for leniency specifically on Yom Kippur.
In summary, we believe that Halacha generally permits use of soap on Yom Kippur in cases in which a basic level of hygiene calls for it. However, one should use simple, not luxurious, soap, and it is even better to dilute it to the point that it has a water-like consistency.[18]