The Gemara in Masechet Shabbat (40) establishes the rule of "Yad Soledet Bo" (literally, "the hand recoils"), which determines the temperature at which liquid is considered "cooked" with regard to the Shabbat prohibition of cooking. Heating water to the temperature of "Yad Soledet Bo" constitutes "cooking" and is thus forbidden on Shabbat. The Gemara explains that this is the temperature at which the water would burn the stomach of an infant on contact.
There is a dispute among the Halachic authorities in identifying the precise temperature of "Yad Soledet Bo." Hacham Ben Sion Abba Shaul (Israel, 1923-1998) was of the opinion that "Yad Soledet Bo" is the temperature of 40 degrees Centigrade, or 104 degrees Fahrenheit. Rav Shlomo Zalman Auerbach (Israel, 1910-1995), however, held that "Yad Soledet Bo" refers to a higher temperature, namely, 45 degrees Centigrade, or 113 degrees Fahrenheit. Others maintain that the temperature is even higher. Hacham Ben Sion noted that the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) defined "Yad Soledet Bo" as the temperature at which water cannot be drunk normally. If so, Hacham Ben Sion wrote, then the temperature of "Yad Soledet Bo" would be 80 degrees Centigrade, which is 176 degrees Fahrenheit. Rav Moshe Feinstein (Russia-New York, 1895-1986) held that "Yad Soledet Bo" is 65 degrees Centigrade, or 149 degrees Fahrenheit.
As for the final Halacha, since we deal here with the Torah violation of cooking on Shabbat, we must follow the stringent opinion. In some cases, this would mean accepting the lowest temperature, while in others it would require following the higher temperature. For example, if a person wants to place some water or other liquid near the hotplate, blech or urn to remove its chill, he may not place it in a spot where it could be heated to a temperature of 104 degrees, the lowest definition of "Yad Soledet Bo." He must ensure to place the water far enough so that it could never reach this level of heat. (It goes without saying that one may not place cold liquid on a hotplate or blech.) Conversely, if a person wants to return hot liquid to the hotplate or blech, he may do so only if the liquid is still at or above the temperature of 170 degrees, the higher definition of "Yad Soledet Bo." If the liquid had dropped below this level, bringing it back to that temperature would constitute cooking (according to some views) and would thus be forbidden. Therefore, one may return a liquid to the hotplate (or blech) only if it still at or above 170 degrees Fahrenheit.
Summary: It is forbidden to heat liquid to the temperature of 104 degrees Fahrenheit on Shabbat. Therefore, if one wants to place cold liquid near a heat source to remove the chill, he must ensure that it is placed far enough that it cannot reach this temperature. A hot liquid that is below the temperature of 170 degrees Fahrenheit may not be reheated back to that temperature.
