Question: My hot water urn has a Shabbat setting, in which the water is heated at a constant level and the switch for boiling the water is disabled. The socket where I plug it in is on a Shabbat clock that is off at night. When it goes on in the morning, the water that has become cold heats back up. Is that permitted?

Answer: Although we accept the opinions among Rishonim that it is forbidden to reheat boiled water that has cooled down (Shulchan Aruch and Rama, Orach Chayim 318:4,15), you would not be considered cooking since this is done automatically.
The question is whether your setup violates the Rabbinical prohibitions of shehiya or chazara. Shehiya, leaving food on the flame from before Shabbat, is sometimes forbidden, out of a concern one will raise the heat. It is permitted if the heat source is covered in a way that reduces its efficiency (Shulchan Aruch, OC 253:1) or (likely) regarding a non-adjustable heat source (Hilchot Shabbat (Eider), p. 340). However, neither lenient factor exists here (one can raise the heat from Shabbat to normal mode). It is usually permitted to use this urn when the water has already been boiled, as further boiling causes unwanted evaporation (see Shulchan Aruch ibid.). Your case could possibly be more problematic since one may desire the extra heat to heat the cold water.
Chazara, returning food on Shabbat that had been removed from the heat, has more stringency, including that it is forbidden on a normal, adjustable heat source even if raising the temperature is detrimental (ibid. 2). Is your case considered chazara, considering that the heat is returned to function by a machine rather than a person? The answer may depend on the reason of the stringency of chazara. Rabbeinu Tam says it is a heightened concern one will raise the heat since the food was returned after time off the flame. The Ran says that returning cooked food to a heat source can be confused with cooking. In this case, Rabbeinu Tam’s reason seems to apply, while the Ran’s does not since you do nothing on Shabbat.
Let us examine discussion about a parallel case. The Pri Megadim (OC, EA 253:41) and the Chazon Ish (OC 37:21) wonder about the permissibility of various cases similar to what the Rama (OC 253:5) allows. A non-Jew may put, on Shabbat morning, cold cooked food near a fireplace, which a non-Jew will be permitted to light due to the great cold, thereby also heating the food. Why are we not concerned that after the fireplace is on, a Jew will stoke the coals? The Pri Megadim suggests that this must rely on the opinion that reheating liquids is permitted, and so too the reheating is not significant enough to prompt one to stoke the coals. The Chazon Ish gives a few possible answers. One is that we treat a case where the food is put down when there is no heat as equivalent to shehiya. This helps since the Chazon Ish claims elsewhere (37:27) that the concern of raising the flame regarding shehiya does not apply to fully cooked food even if it is now cold. On the other hand, reheating cooled water may be worse than reheating other cooked foods (Orchot Shabbat 2:(11)). There is further room for leniency considering that Shabbat started with the urn operating and there was no action since then (see Am Mordechai, Shabbat, p. 51). Still, the Shemirat Shabbat K’hilchata (1:40) is stringent when the water has cooled off totally, and the Orchot Shabbat (2:(49)) is uncertain.
Your urn has a feature that provides further grounds for leniency – when the Shabbat mode is on, one cannot raise the heat. This is similar, in some ways, to one who seals an oven where food is heating, which is permitted even though the seal can be removed (Shabbat 18b). It is unclear if the Shabbat-mode button that is deactivated by a simple press is sufficient deterrent (see cases in Orchot Shabbat 2:18-19). It is also unclear if this leniency applies when elements of chazara exist (see ibid. 55). However, combining this factor along with the aforementioned grounds for leniency, it is not difficult to justify leniency.