Showering on Yom Tov
Heating up water on Yom Tov
In the Mishna (Beitzah 2:5), Bet Shamai says that one may only heat up water for washing one’s feet to a temperature that one would be able to drink, while Bet Hillel permits heating up the water even as hot as needed to bathe. Bet Hillel permits 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) (Ran Beitzah 11a). Since we always hold like Bet Hillel (Eiruvin 6b) it should be permitted to heat up water for washing. However, the Gemara Beitzah 21b writes that even Bet Hillel agrees that it’s forbidden to heat up water one’s entire body. There are two major explanations as to why it’s forbidden to heat up water for the rest of the body.
Tosfot (21b D”H 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 Ketubot 7a). However, the Rambam (Yom Tov 1:16) holds that the only prohibition involved is the rabbinic prohibition to bathe in a bathhouse. [This is also the opinion of the Ritva Beitzah 21b D”H Matiten.]
Shulchan Aruch 511:2 rules like Tosfot that it is forbidden to heat up water on Yom Tov to wash anything more than one’s hands. (It’s evident that Shulchan Aruch holds like Tosfot from Bet Yosef 511:2. So writes Chazon Ovadyah (Yom Tov pg 41).) It’s implied from Biur Halacha 511:2 D”H Yadav 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 holds that there’s a rabbinic prohibition not to bathe in water heated before Yom Tov just like there’s a rabbinic prohibition regarding water heated before Shabbat. 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 forbid washing one’s whole body in water that was heated before Yom Tov. However, the Sephardic minhag is to follow Shulchan Aruch. (Sh”t Mayim Chayim 1:29, Chazon Ovadyah (Yom Tov pg 41)).
- 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 not in a bathhouse. One may shower on Yom Tov Sheni with water heated by an electric heater on Yom Tov Rishon. 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. 
- 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 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. 
- "Changes in Sociology or Technology and Jewish Law Responses to Them: The Cases of Showering or Smoking on Yom Tov" is a good 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 p41,157)
- Mishna Brurah 551:9 and 18 rules 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. Shemirat Shabbat KeHilchata 14:7 (in old and new edition) rules like Mishna Brurah. [In the footnote he discusses the topic at length about shows that there is some reason to be lenient but nonetheless doesn't rule that way.] 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.
- This is the opinion of Rabbi Aryeh Lebowitz mostly quoting Rav Hershel Schachter. Here's the background to why he is lenient. (1) Magan Avraham 511:5, Aruch HaShulchan 511:5, and Biur Halacha 511 D”H 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. (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 in 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.