On Shabbat, one may not open a hot water faucet. Although the water in the electric boiler was already heated before the tap was opened, the action of turning on the faucet causes cold water to enter and replenish the boiler. This new cold water is, in turn, heated by the hot water remaining in the boiler. This process of heating the new water constitutes cooking.
It is still prohibited even if the boiler is switched off. The hot water is considered the heating element, even if there is no fire. The Shulhan Aruch (318:3) states that there is no difference between cooking with fire ("Ur") and cooking via something else which was heated by fire ("Toldot HaUr"). Both actions are Torah prohibitions.
Moreover, the fact that one has no interest in the cold water which is heated is irrelevant. This action is called a "Pesik Resheh"-an inevitable outcome of turning on the faucet. In cases of Torah prohibitions like this, the fact that one has no interest or benefit from the inevitable outcome of cooking does not make the action permissible.
The Poskim discuss what to do in the event one accidentally turned on the hot water faucet on Shabbat. Intuitively, one might think the best thing is to quickly turn it off. However, Hacham David, in his Halacha Berurah, and Hacham Gidon point out that, on the contrary, one should leave it running. Closing the faucet would prevent new cold water from entering the boiler and thereby augment the cooking of the cold water that entered. This constitutes a type of prohibited cooking on Shabbat. A non-Jew may close the faucet, as there is no issue of "Pesik Resheh" with regard to a non-Jew.
Thus, just as it is prohibited to turn on the hot water faucet, it is also prohibited to close it.
SUMMARY
It is prohibited to turn on or turn off a hot water faucet on Shabbat.
