By Rabbi Dovid Ostroff
Is one permitted to take a hot shower on Shabbos?
When discussing the issue of taking hot showers on Shabbos, two factors must be taken into consideration. The first deals with use of a hot water system, which incidentally would not make a difference whether it is used for washing one’s hands, washing dishes or a person’s body. The second deals with washing one’s body with hot water.
Opening the hot water faucet on Shabbos can easily involve an issur d’oraisso – a biblical violation, depending on the type of hot water system. Without entering that complicated subject, the general halachic rule is that one must not open the hot water tap on Shabbos, regardless of which system is in operation.
The second issue is the focal point of our discussion. For argument sake, and for want of avoiding the issue of using a hot water system on Shabbos, we will talk about taking a hot bath in a bathtub that was filled with hot water before Shabbos.
Seemingly there can be nothing wrong. However the gemora Shabbos 39b says that one may not wash one’s entire body with hot water on Shabbos, regardless of when the water was heated.
What is the reason for this prohibition?
Chazal instituted this g’zeira because if one would be permitted to bathe in hot water, there is fear that one would heat water for that purpose, which of course involves an issur d’oraisso. 
Why do you say that one may not wash one’s entire body?
Chazal only prohibited the washing of the greater part of one’s body with water that was heated before Shabbos. One may wash one’s face hands and feet (or limited areas of one’s body for that matter) with water that was heated before Shabbos.  As stated, we are avoiding the issue of opening a hot water tap and are therefore referring to hot water in a bath or in a basin from before Shabbos. Water on the Shabbos hot-plate (obviously from before Shabbos) is considered heated before Shabbos, not on Shabbos.
Is the halacha different for a person who is ill?
A person who has a minor illness and would benefit from washing his body with hot water that was heated before Shabbos may do so.  The reason is because the entire prohibition is an issur d’rabanan and Chazal did not institute this g’zeira when illness is involved.
What is the halacha with regards to water that was heated on Shabbos?
One may not wash even one limb with water that was heated on Shabbos. Rav Shlomo Zalman Auerbach ztz”l defines water heated on Shabbos as water that was actively placed on Shabbos in a place where it heated up. However, water that was placed on a hot-plate before Shabbos and heated up on Shabbos is called water heated before Shabbos.
May one warm one’s wet body next to a fire?
First we must state that the water is not heated to the degree of yad soledes bo  because that would be cooking the water. Second, we must understand and analyze the problem. Is this heated water considered as water heated on Shabbos, thus prohibiting even the warming of one’s wet hands, or is it not considered heated on Shabbos?
We find a machlokes regarding this matter.
The Rambam holds  that when one wishes to wash one’s body with hot water one does not normally wet one’s body and stand next to a fire, consequently he classifies the water as heated ‘before Shabbos’. Accordingly one may warm one’s hands (provided that the water does not reach yad soledes bo) next to a fire but not one’s wet body.
The Rosh  however classifies the water as heated on Shabbos and thus one may not warm even one’s wet hands next to a fire on Shabbos. Accordingly one may not warm wet hands on the heating unit on Shabbos. The Bi’ur Halacha adds that one should take care not to warm wet hands on a hot wall that radiates heat next to a heater.
How should one mix hot water and cold in order to wash one’s hands or face?
One of the methods is to take hot water from an electric Shabbos urn or to use some water from the kettle that is on the Shabbos hot plate or blech. This water is heated before Shabbos and may be used for this purpose.
The problem is that this water is too hot to bathe with, and for practical reasons it must be mixed with cold water. Mixing hot and cold water can result in ‘cooking’ the cold water and therefore must be mixed in a permitted manner.
Two methods are feasible and permitted:
1) Pour hot water from the urn etc. into a dry vessel and add cold water to cool it. Since the vessel the hot water is now in is called a k’li sheini, one may add ‘uncooked’ water to a k’li sheini and it does not ‘cook’ in that k’li. 
2) Pour hot water from the urn etc. onto cold water making sure that only a small amount of hot water is poured thus ensuring that the mixture does not reach the temperature of yad soledes bo. 
1] Siman 322:5 and M”B 17.
 Siman 326:1 and M”B 1.
 Bi’ur Halacha in the name of R’ Akiva Eiger.
 40º-45ºC or 104º – 113ºF.
 As quoted by the Mechaber in siman 326:4.
 As quoted by the Mechaber in siman 326:5, and explained in M”B 17.
 Siman 318:13. See also the SS”K 1:53.
 See the SS”K 1:51 based on the Bi’ur Halacha in siman 318:12. Yad soledes bo: 40º-45ºC or 104º – 113º