1) BATHING ON SHABBOS AND YOM TOV
QUESTION: One is forbidden to bathe his entire body with hot water on Shabbos and Yom Tov because of a Gezeirah that was enacted in order to prevent the bath attendants from heating water on Shabbos.
Why, though, does the Gezeirah against bathing apply on Yom Tov? On Yom Tov one is permitted (even l'Chatchilah) to heat water for bathing, as the Mishnah in Beitzah (21b, and as quoted here on 39b) teaches!
(a) TOSFOS (39b, DH u'Beis Hillel) and the Yerushalmi answer that since rinsing one's entire body is an indulgence that is "Eino Shaveh l'Chol Nefesh" -- it is not a pleasurable activity commonly done by everyone (in those days), the allowance to do a Melachah for such a purpose on Yom Tov does not apply, and it remains Asur mid'Oraisa.
(b) The RAMBAN, RE'AH, and CHIDUSHEI HA'RAN argue that washing one's entire body is certainly a normal, pleasurable activity done by everyone, and heating water for that purpose is permissible. The reason why one may not bathe in hot water on Yom Tov is not because of the same decree that forbids bathing on Shabbos, but rather because one might come to transgress any one of several other Melachos that bathing may entail, such as smoothing the floor of the bath or squeezing out a towel.
2) HALACHAH: WASHING ONE'S ENTIRE BODY ON SHABBOS
The Gemara concludes that the Halachah follows the view of Rebbi Yehudah, who says that one is permitted to wash with cold water, but not with hot water, on Shabbos. However, the custom is not to wash even with cold water, because there are many prohibitions that one might accidentally transgress (such as squeezing the towel or one's hair).
One who must immerse in a Mikvah may do so on Shabbos in an unheated Mikvah, as long as one is careful not to squeeze the towel or one's hair, or carry something in Reshus ha'Rabim to the Mikvah. (SHULCHAN ARUCH OC 326:8 and BI'UR HALACHAH there)
3) PLACING A KETTLE OF HOT WATER ON ONE'S STOMACH
OPINIONS: The Gemara states that a person should not place a kettle of hot water on his stomach on Shabbos. What is the reason for this prohibition on Shabbos?
(a) RASHI says that it is prohibited because the hot water may spill over onto the person and it will be as if he is washing his body on Shabbos with hot water. (This apparently is a Gezeirah for a Gezeirah. It is evident from Rashi's words that washing the torso is equivalent to washing one's entire body, which is prohibited on Shabbos.)
(b) TOSFOS disagrees and says that the Gemara is discussing a case in which there is a towel on one's stomach underneath the kettle. The Rabanan were concerned that the water would spill over from the kettle and one would squeeze out the towel.
(c) Tosfos gives an alternative reason and says that it is prohibited to place hot water on one's stomach when it is done for medicinal purposes. This act is included in the decree of the Rabanan that prohibits medical acts on Shabbos.