May an electric water urn be used for Shabbos?
The water is heated based on a thermostat: Some Poskim rule a thermostat based electric water urn may not be used on Shabbos as the amount of water that is in the pot effects when the heat will turn on and off. It is thus forbidden to remove water from it on Shabbos [even when the heat is on, as nevertheless this will cause the heat to turn off earlier than expected, and certainly when it is off, as this will cause it to turn on]. Other Poskim however rule it is permitted to be used.
Shabbos mode-The water is not heated based on thermostat: A non-thermostat based electric water urn may be used on Shabbos. Such an urn is commonly known as a Shabbos mode urn, which is an urn that has a Shabbos mode option that makes the heat turn on periodically, and not based on temperature.
 Shabbos Kehalacha Vol. 1 p. 240
 Minchas Yitzchak 5/91; Orchos Shabbos 2/26; Shabbos Kehalacha ibid
 Orchos Shabbos ibid
 SSH”K 1/40 [46 new edition] in name of RSZ”A; Maor Hashabbos 1/12 in letter of Rav SZ”A; See Igros Moshe 4/74 Bishul in which a number of Teshuvos rule that it may be used and no mention is made of any prohibition; Piskeiy Teshuvos 254 footnote 45 one who is lenient has upon whom to rely.
 The reason: As it is a case of Germa Delo Nicha Lei and is not a full Pesik Reishei. [Rav SZ”A ibid]
 Admur 254/14
 This refers to all urns that have adjustable modes, or a switch to turn on/off.
 Igros Moshe 4/74 Bishul 24; Shabbos Kehalacha Vol. 1 pages 317-319; See Ketzos Hashulchan 71 footnote 1; Piskeiy Teshuvos 254/6
 Admur 253/1 “if by the time the night [of Shabbos] begins the dish is still not cooked to the point of Ben Drusaiy.” M”A 253/26; Peri Megadim 253 A”A 26; Rav Akivah Eiger 253; Shabbos Kehalacha Vol. 1 pages 313-314;
 SSH”K 1/40; Piskeiy Teshuvos 318/61; See Minchas Yitzchak 3/137; 5/91; 6/21; Az Nidbaru 7/21; Shevet Hakehasi 3/109; 4/108
 The reason: This is due to suspicion that the water may completely finish and one may come to add more water in order to prevent the urn from burning up. [See M”B 318/117]
 Minchas Yitzchak ibid; Piskeiy Teshuvos ibid
 See Piskeiy Teshuvos 318 footnote 254
 Rav Moshe Yadler in Maor Hashabbos end of volume 1 and Minchas Yitzchak 10/28 [unlike his ruling in 6/21]; Minchas Yitzchak ibid according to those who hold there is a prohibition of cooking by water that was not boiled before Shabbos; So rules Rav Asher Lemel Cohen; and the Machon for Halachic technology
 The reason: The reason for this is because they suspect the water in the tube does not become Yad Soledes and when one releases water from the urn some of that water in the tube consequently enters the urn of hot water and gets cooked. Furthermore, even if it did reach Yad Soledes, some Poskim rule that so long as the water did not rach boiling point it is not fully cooked and retains a cokking prohibition, and hence when one now removes water from the urn it causes the uncooked water to mix in with the cooked water and further cook. [Igleiy Tal in Hashmatos, retracting from his ruling in the “Ofeh” 8; Tehila Ledavid 318/17 and Rav SZ”A in SSH”K 1 footnote 96 leave this matter in question; Sheivet Haleivi 7/42 and Minchas Yitzchak 10/28 lean to be stringent]
 Orchos Shabbos 2/30; Minchas Shlomo 34/23; Shalmei Yehuda 53 in name of Rav Elyashiv; Az Nidbaru 9/14; Shevet Hakehasi 4/108; Or Letyziyon 2/30-10; See Minchas Yitzchak ibid that it is possible the water reaches Yad Soledes and hence there is no prohibition according to those who rule that Yad Soledes is already considered cooked;
 The reason: As the water in the tube certainly reaches above Yad Soledes [Piskeiy Teshuvos ibid says it reaches between 55-70 celsius] and many Poskim rule that once water has reached Yad Soledes it is considered cooked and no longer has a cooking prohibition to further heat. [Igros Moshe 4/74 based on Admur 318/9; Az Nidbaru 9/14 seemingly retracting his ruling in Bris Olam; Olas Shabbos 318/12; Shevisas Shabbos 18; Iglei Tal Ofeh 8] Alternatively, all the water in the urn and valve get mixed during the boiling process, and hence according to all everything is already pre-cooked and no longer has a cooking prohibition so long as it is still warm. [Orchos Shabbos ibid] Alternatively, it is permitted being it is a Safek Gerama and not Niacha Lei. [Minchas Shlomo]