The Halacha permits instructing a non-Jew to do an act known as a "Písik Resheh"- a permitted action that causes an inevitable automatic violation of Shabbat. For example, it is permitted to tell a non-Jew to open a refrigerator door, even though, inevitably, the light will turn on.
This principle can be applied to taking hot water from an urn which has an automatic refill function. That is, whenever the mechanism is used to draw hot water into a cup, cold water is automatically streamed into the heating chamber to be heated. Clearly, a Jew is prohibited from using such an urn, since every time he takes hot-water, he is doing a "P'sik Resheh" of cooking the new water. However, based on the aforementioned principle, it is permissible to tell the non-Jew to pour a cup of hot water.
Similarly, it is forbidden to turn on the hot water in a sink on Shabbat. Even if all the water currently in the tank is heated from before Shabbat, nevertheless, as hot water is drawn from the sink, new cold water is pumped into the tank and is heated on Shabbat. The fact that this process is not seen makes absolutely no difference. Nobody would think that is permissible to turn on a light on Shabbat with their eyes closed! Since this is a "P'sik Resheh," one may instruct a non-Jew to turn on the hot water faucet. If the cold water intake valve was shut before Shabbat, it is permitted even for a Jew to turn on the hot water.
This leniency can also be applied to Yom Tob. Although on Yom Tob, as opposed to Shabbat, it is permitted to wash the entire body in hot water, nevertheless, it is still prohibited to heat the water for that purpose. Therefore, one may ask a non-Jew to turn on the hot water of a shower on Yom Tob, in order to wash oneself.
A Jew is prohibited from turning on the hot water faucet on Shabbat, but he may instruct a non-Jew to do so.