Showering on Yom Tov
Heating up water on Yom Tov
In the Mishna (Beitzah 2:5), Bet Shamai say that one may only heat up water for washing one’s feet and only to a temperature that one would be able to drink, while Bet Hillel permit heating up the water even as hot as needed to bathe. The Rif and Tosfot (Beitzah 21b) stipulate that Beit Hillel only permitted heating water on Yom Tov for one's hands and feet, but not for one's entire body.
The Rishonim's question
The Ran (Beitzah 11a) explains that Bet Hillel's opinion is based on the concept of Mitoch (since it’s permitted to cook for food on Yom Tov, it’s also permitted to heat up water for bathing). This concept of mitoch is a subject of dispute between Beit Hillel and Beit Shammai in an earlier Mishna. The Mishna (Beitzah 1:5) records a dispute where Bet Hillel permitted carrying a child, Lulav, or sefer torah in a public domain on Yom Tov, and Bet Shamai prohibited. The Gemara (Beitza 12a) explains that this dispute is based on the concept of Mitoch, Beit Hillel accept it, while Beit Shammai do not. Since we always hold like Bet Hillel (Eiruvin 6b), Shulchan Aruch Orach Chaim 518:1 rules that we hold like Beit Hillel that there is a concept of mitoch. Seemingly, then it should be permitted to heat up water for washing one's entire body based on the concept of mitoch.
Tosfot (Beitzah 21b s.v. Lo Yicham) writes that Bet Hillel only allows heating up water to a high temperature regarding one’s hands and feet because that’s a pleasure enjoyed by everyone, however, for the entire body it’s forbidden (Biblically) because it’s not considered something which is a pleasure enjoyed by everyone. This is similar to the Gemara Ketubot 7a which stipulates that a pleasure which is only enjoyed by wealthy people isn't considered Ochel Nefesh. A number of Rishonim agree with this explanation.
The Rambam (Yom Tov 1:16), however, holds that the only prohibition involved is the rabbinic prohibition to bathe in a bathhouse. A number of rishonim agree.The Ramban (Shabbat 40a s.v. Ha Ditnan) explains that even though seemingly this rabbinic prohibition should not be extended to Yom Tov because it is permitted to engage in those melachot for Yom Tov, nonetheless, since some Halachic concerns still apply to bathing on Yom Tov, such as squeezing water from one’s hair or towel, the prohibition applies to Yom Tov as well.
Which approach is accepted as halacha?
The Shulchan Aruch 511:2 seems to rule like the Rambam, while the Rama follows the opinion of Tosfot. For a clarification of this point as well as the opinion of other major poskim, see the footnote. The Beiur Halacha 511:2 s.v. Yadav implies that what is considered a pleasure enjoyed by everyone depends on the time and place. (Sh”t Mayim Chaim 1:29 supports this idea.) 
Showering with water heated up before Yom Tov
Tosfot who holds that there is a biblical prohibition to heat up water on Yom Tov, also holds that there’s a rabbinic prohibition to bathe in water heated before Yom Tov just like there’s a rabbinic prohibition regarding water heated before Shabbat. Based on this, the Noda Biyehuda OC 24 and Chacham Tzvi 11 forbid immersing in a heated mikveh, even if the water was heated before Yom Tov but they did permit using lukewarm water. Teshuvot Divrei Chaim OC 2:26 says that the minhag is to allow women to immerse even on Shabbat and Yom Tov in a heated Mikvah. Rabbi Akiva Eiger on Shulchan Aruch 307:5, quoted by Bi’ur Halacha 326:1 "Bimayim" permits bathing in hot water even on Shabbat because if not the women would be in great discomfort, and the gezeira wasn't made on situations like those. However, the Rambam holds that it’s permissible to wash one’s whole body with water that was heated up before Yom Tov as long as it’s done outside a bathhouse because there was a rabbinic prohibition not to bathe in a bathhouse on Yom Tov (Bet Yosef 511:2).
Shulchan Aruch 511:2 rules like the Rambam and the Rama rules like Tosfot. Mishna Brurah 551:19 writes that Ashkenazim shouldn’t change the minhag to refrain from washing one’s whole body in water that was heated before Yom Tov. However, Chazon Ovadyah (Yom Tov pg 41) writes that the Sephardic minhag is to follow Shulchan Aruch. Sh”t Mayim Chayim 1:29 agrees.
One may not shower on Yom Tov with water heated on Yom Tov, however, one may shower with water heated before Yom Tov as long as it is not in a bathhouse. One may shower on Yom Tov Sheni with water heated by an electric heater on Yom Tov Rishon, provided there is no incoming cold water that will be heated as it enters the tank. Water that was heated in a solar boiler (common in Israel) is considered like water that was heated before Yom Tov and one could shower one’s whole body in them on Yom Tov. 
According to a ruling of Rav Yitzchak Yosef Shlita, in the Diaspora, when Yom Tov falls out on Thursday and Friday and leads into Shabbat, colloquially known as a Three Day Yom Tov, it is permissible to take a shower on Yom Tov as long as the water in the boiler was boiling before Yom Tov even if there is incoming water that will be heated as it enters the boiler.  In any event, this ruling does not address the issues of plucking out hairs or squeezing out hair, which are forbidden.
Ashkenazik Many major authorities forbid taking a shower with hot water whether the water was heated before Yom Tov or on Yom Tov. However, one may wash one's body one limb at a time with water heated before Yom Tov and one may heat up water on Yom Tov to wash one’s face hands and feet. However, some authorities are lenient and hold that it's preferable to shower on Yom Tov with lukewarm water but it's even permitted to shower with hot water, however, one should not use very hot water. Nonetheless, it's imperative to be careful about certain concerns:
- (1) that one may not squeeze water out of one's hair, either to dry it or to rub in shampoo and so one should put the shampoo in before wetting one's hair.
- (2) One may not use regular bar soap based on Mishna Brurah 326:30 but one should use liquid soap, preferably one that is a thin liquid.
- (3) Some forbid using the towel to dry one's hair but agree that it's permissible to dry one's body as long as the towel isn't saturated to the point that one is squeezing out water. 
- Rabbi Ari Enkin on hirhurim.com
- Rabbi Jachter on koltorah.org
- Halachically Speaking
- Rabbi Aryeh Lebowitz on yutorah.org
- Rav Hershel Schachter in Reishit Bikkurim (p. 5)
- The idea of Tosfot Beitzah is also found in Tosfot (Shabbat 39b s.v. VeBeit Hillel), Rosh (Shabbat 3:7), Meiri (Beitzah 21b s.v. Amar HaMeiri HaMishna HaReviyit) ,and Piskei Rid (Beitzah 21b according to the Gilyon of the Ketav Yad).
- The Ritva Beitzah (21b s.v. Matiten) agrees with the Rambam that the prohibition to heat up water on Yom Tov is only rabbinic in nature. The Shitah Mekubeset quotes this.
- Proofs that the halacha follows Tosfot: The Beit Yosef 511:2 simply quotes Tosfot and does not quote the reason of the Rambam. Seemingly, he holds like the Tosfot. Even though, he quotes the Rif, who says that one may not heat up water for a complete body shower, the Rif doesn't explicitly state that he agrees with the Rambam's explanation. Mishna Brurah 511:10 quotes the opinion of Tosfot and only mentions the explanation of the Rambam in the Shaar HaTziyun (511:8). Chazon Ovadyah (Yom Tov pg 41) also explains the prohibition to heat up water on Yom Tov for a full-body shower in accordance with Tosfot.
- Proof that Shulchan Aruch follows the Rambam: The Ran (Shabbat 18b and Beitzah 11b) writes that since the Rif and Rambam hold that heating up water on Yom Tov for a full-body shower is onlyrabbinic, it is permitted to take a shower if the water was heated up before Yom Tov. According to Tosfot, who hold that heating up water on Yom Tov is a biblical prohibition, it would be prohibited to take a shower with hot water even if it was heated before Yom Tov. Beit Yosef 511:2 quotes the Ran. Shulchan Aruch 511:2 rules that it is permitted to bathe one's entire body in hot water which was heated before Yom Tov, while the Rama rules that it is forbidden. Using the logic of the Ran, Shulchan Aruch follows the opinion of the Rambam, whereas the Rama follows the opinion of the Tosfot.
- Assuming, as did the Ran, that the Rif holds like the Rambam, it is very reasonable to believe Shulchan Aruch follows the Rambam, since two of the three major pillars of halacha agree to that opinion. See Rav Yosef Karo's introduction to the Beit Yosef s.v. VeLeChen. See, however, the Meiri (Beitzah 21b s.v. Amar HaMeiri HaMishna HaReviyit) who explains like Tosfot but also rules like the Rif that one may completely bathe in water which was heated before Yom Tov. According to the Meiri, these two aren't necessarily related and as such, the proof as to the ruling of Shulchan Aruch isn't ironclad.
- Proofs that we're strict for both: The Kaf HaChaim 511:13 quotes both opinions and doesn't rule definitively. The Shaar HaTziyun details a practical difference between the Rambam and Tosfot but doesn't rule either way.
- "Changes in Sociology or Technology and Jewish Law Responses to Them: The Cases of Showering or Smoking on Yom Tov" is an RJJ article by Avi Wagner and R’ Broyde on this topic. (R’ Broyde states aware of only one posek who says “Shaave” never changes).
- Rav Ovadyah Yosef (Chazon Ovadya Yom Tov p. 41 and p. 157), Yalkut Yosef Shabbat Volume 4 Siman 326 page 58
- Rabbi Gavriel Elbaz (author of the English HalachaYomit site) as well as a few other Rabbis sent a question to Rav Yitzchak Yosef Shlita as to whether the lenient position Rav Ovadia Yosef took regarding a solar boiler could be applied to an electric boiler. The final ruling of Rav Yitzchak Yosef can be found on HalachaYomit.co.il in Hebrew as well as HalachaYomit.co.il in English. See, however, another article on HalachaYomit.co.il.
- The letter by Rabbi Elbaz included the following points. Please note that the following variables individually may not be accepted by the halacha, however, it is together that they formed a basis for a lenient position.
- Facts of the case: Under the boiler is a gas fueled flame that is constantly lit to keep the water hot. Inside the boiler is a thermostat to measure the temperature of the water. When water is removed from the boiler, immediately cold water enters the boiler. At that time, either the cold water is heated immediately by the hot water in the boiler or because a lot of cold water entered, the temperature drops and the fire below the boiler is raised automatically.
- Perhaps the entire prohibition of heating up water, according to Shulchan Aruch, is only rabbinic (see above).
- Perhaps the new water that enters and is heated is only considered a pesik reisha de'lo nicha leh. See Chazon Ovadia (Yom Tov p. 41) and Yabia Omer 4:34:35. Even if it is a pesik reisha denicha leh, still some permit a pesik reisha for a rabbinic prohibition. See Leviyat Chen (Siman 314 #39).
- There is a minority opinion that considers the water that entered as grama. See Chazon Ovadia (Shabbat v. 4 p. 405).
- If it can be established that the water was heated up in a permitted fashion, then there is a discussion whether it is permitted to take a full body shower in those waters. See Shulchan Aruch 326 and 511, Halichot Olam (v. 4, p. 207), and Chazon Ovadia (Shabbat v. 6, p. 76-8).
- Lastly, perhaps for someone who is in pain there is room to be lenient. See Rabbi Akiva Eiger (307 and 326).
- This is the opinion of Rabbi Aryeh Lebowitz mostly quoting Rav Hershel Schachter. This lenient opinion is based on a discussion whether and to what extent halacha considers a pleasure enjoyed by most people to change according to time.
- (1) Magen Avraham 511:5, Aruch HaShulchan 511:5, and Beiur Halacha 511 s.v. Yadav hold that when considering what is a pleasure enjoyed by all one must take into consideration the contemporary custom during the week.
- (2) Rabbi Aryeh Lebowitz (New York, 2010) brings a proof from the above sources that since nowadays it’s more common to shower daily or at least once every two days heating up water for that purpose should be permissible. Rabbi Lebowitz quotes Rav Hershel Schachter who permits showering in lukewarm water. It should be noted, however, that Rav Hershel Schachter's pesak as recorded in Reishit Bikkurim (Shavuot 5772) is that it is forbidden to shower in hot or warm water, but only in water that's cold but not uncomfortably cold. In the footnote he explicitly writes that lukewarm water is forbidden. This same position is confirmed in a shiur on yutorah.org titled "Hilchos Pesach and Yom Tov" (min 30) which was given before Pesach 2013. (3) Sh”t Mayim Chayim 1:29 (Rav Chaim Dovid HaLevi, Tel Aviv, 1991) agrees with the above proof, however, he disagrees with the assumption that nowadays the common custom is to shower daily and it’s not sufficient that it’s the custom to shower regularly (even once in two days). [He does admit that if he were in a very hot climate and it was actually the custom for everyone to take a shower every single morning then he’d permit.] (This is similar to the position taken by the Chazon Yechezkel Beitzah 2:7 but Shemirat Shabbat Kehilchata 21 note 21 questions it.)
- (4) However, Sh”t Bear Moshe 8:158-9 (Rav Moshe Stern, Brooklyn, NY, 1987) argues that the above sources aren’t a proof because in all those cases the contemporary custom was only considered in order to create a stringency beyond that which that was ruled in Shulchan Aruch. Therefore, Bear Moshe is unwilling to consider permitting heating water to shower on Yom Tov.
- (5) Mishna Brurah 551:9 and 18 rules (like the Rama) that one may not take a shower a full body shower with hot water on Yom Tov whether the water was heated on Yom Tov or before, however, one may wash one's whole body part by part with water heated before Yom Tov and one may heat up water on Yom Tov to wash one's hands, feet, and face.
- (6) In conclusion whether or not this definition of "a pleasurable activity enjoyed by all" in halacha can change most poskim are unwilling to rule leniently and rule in accordance with the Mishna Brurah even nowadays. This is the opinion of Shemirat Shabbat KeHilchata 14:7 (in old and new edition), Halachically Speaking quoting Horav Yisroel Belsky, Rivevot Ephraim i6:265, 8:248:1, Sh"t Tzitz Eliezer 6:20, 11:64, Avnei Yushfei 3:55 quoting Rav Elyashiv. [In the footnote (#21) of Shemirat Shabbat Kehilchata he discusses the topic at length and shows that there is some reason to be lenient but nonetheless doesn't rule that way.]
- (7) Rabbi Simcha Bunim Cohen in his book "The Laws of Yom Tov" basically agrees to all of the above and adds that it's customary to refrain from even a cold shower on Yom Tov unless there it is a case of discomfort and that it's permitted to heat up water on Yom Tov for the purpose of washing a minor part of the body. Lastly, Rav Simcha Bunim Cohen adds that washing any part of the body in a shower or bathtub is forbidden unless one is partly clothed. Rav Chaim Jachter on koltorah.org points out that one should not distinguish in this regard between the first and second day of yom tov simply because the second day is rabbinic, as the Aruch Hashulchan 511:11 says this is degrading to Yom tov sheni.