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)).
- Since today’s bathrooms are different than ancient bathhouses, one may
- Shower on Yom Tov with water heated before Yom Tov as long as not in a bathhouse
- Shower on Yom Tov Sheni with water heated by an electric heater on Yom Tov Rishon 
- Mishna Berurah  since whole body is not “Shaave Lakol”, it’s prohibited.
- Shmirat Shabbat Kihilchitah 
- Washing the whole or most of one’s body at once in warm water is forbidden on Yom Tov even if the water was heated before Yom Tov
- Water Which was heated before Yom tov may be used to wash parts or even the whole of one’s body part by part
- Water may be heated on Yom Tov itself to wash one’s face hands and feet.
- ponders “Shaave" at length , especially if one is very dirty.
- even if not "Shaave” the way our pipes work, the hot water will also go to the kitchen sink to wash hands “making tosafot’s fear remote”
- Rabbi Simcha Bunim Cohen 
- It is prohibited to bathe or shower one’s entire body or the majority of the body in warm water on Yom Tov, even if the water was heated up the day before
- It is customary to refrain even form bathing or showering with cold water on Yom Tov
- In cases of discomfort (e.g. a person perspiring heavily) it is permitted to bathe or shower with cold water
- It is permitted to wash the entire body one limb at a time with cold water heated before yom tov. However it is forbidden to heat water or use hot tap water for this purpose
- it is permitted to heat up water for the purpose of washing a minor par of the boy
- washing any part of the body in a shower or bathtub is forbidden unless one is partly clothed
- Rabbi Simcha Bunim Cohen 
- 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. 
- It's not permissible to heat up water in order to bathe in the mikveh but one may heat it a little to take the chill out of the 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 511:18
- This is discussed in his book "The Laws of Yom Tov".
- Chazon Ovadyah (Yom Tov pg 41)
- Mishna Brurah 511:25