One may not bathe in, wash or shower with water heated on Shabbat. Whether one may perform these activities with water heated before Shabbat is debated by Rav, who maintains that one may wash one’s entire body, limb by limb, in such water, and Shmuel, who maintains one may only wash one’s face, hands and feet in such water.
As with shehiyah and chazarah, the rabbis restricted the ways one may use such water for fear that a person might reheat water on Shabbat – an act that is forbidden according to all. Indeed, this fear was grounded in experience for, as the Talmud relates, originally one was allowed to bathe in water heated before Shabbat, but later the rabbis forbade it because the bathhouse attendants would heat the baths on Shabbat and claim they’d heated them before Shabbat.
The Shulchan Aruch determines the halacha in accordance with Shmuel’s opinion whereas the Rema determines it somewhere between Rav and Shmuel’s opinions – in other words, that one may wash one’s other limbs too in such water but not one’s entire body.
May one use hot water from the tap on Shabbat? This depends on whether the water coming out of the tap was heated on Shabbat, in which case it would be forbidden, or whether it was heated before Shabbat, in which case it would be permitted, subject to the restrictions described above.
But even if the initial water coming out of the hot water tap had been heated on Erev Shabbat, opening it on Shabbat would be prohibited if this would inevitably cause cold water to re-enter the boiler in the hot water’s place and be heated on Shabbat. Accordingly, Reb Moshe Feinstein permits using water from the hot water tap on Shabbat if one turns off the boiler approximately two hours before Shabbat. Turning off the boiler guarantees that cold water re-entering the boiler will not be heated on Shabbat, neither by the fire nor by the hot water in the boiler, which by Shabbat would have cooled down to below the temperature of yad soledet bo (40C to 70C). This solution may be practical in a one-family home but not in an apartment house.
One may use the mikveh on Shabbat. It is a generally accepted custom to abstain from swimming on Shabbat.
It should be noted that the Mishnah Berurah quotes halachic sources that permit washing one’s entire body in water heated before Shabbat if not doing so would cause one to suffer on Shabbat.