The Shulhan Aruch (Orah Haim 326:1) rules that although one may not bathe the majority of his body in warm water on Shabbat, it is permissible to wash one’s hands, face and legs with warm water that was heated before Shabbat. If, however, water was heated by a Jew on Shabbat in violation of Halacha, then it may not be used for any purpose on Shabbat.
The question arises as to the status of hot water which was heated on Shabbat in a permissible fashion. One common example is an urn plugged into a timer which one set before Shabbat. If the timer activates the urn on Shabbat and warms the water, would it be permissible to use the heated water to wash one’s hands or feet? Another example is the solar-powered boilers ("Dud Shemesh") which are very common in Israel. Many people in Israel have a boiler hooked up to a solar panel on their rooftops, where water is heated by the sun, without the individual doing anything to heat the water. Would it be permissible on Shabbat to use water which was heated on Shabbat by the sun in a solar-power boiler?
Hacham Bension Abba Shaul (Israel, 1924-1998) ruled stringently in this regard, based on the Shulhan Aruch’s ruling (Orah Haim 326:5) forbidding drying one’s wet hands on Shabbat by placing them near a fire. According to Hacham Bension, this is because the water is warmed by the fire, and then one ends up "washing" his hands with water heated on Shabbat. Hacham Ovadia Yosef, however, both in Halichot Olam and in Hazon Ovadia (vol. 6), disagrees, noting that numerous Halachic authorities, including the Gaon of Vilna (Rav Eliyahu of Vilna, 1720-1797), permit using water which was heated permissibly on Shabbat. Therefore, in the situations described above, where water was heated on Shabbat by an urn activated by a timer, or by a solar-powered boiler, one may use the hot water, provided, of course, that he does not bathe the majority of his body.
Summary: Although one may not bathe on Shabbat, it is permissible to wash one’s hands, face and feet with water that was heated before Shabbat, or with water which was heated permissibly on Shabbat, such as in an urn activated on Shabbat by a timer, or a solar-powered boiler.