Gemara

(Rava): After dark, Hatmanah (wrapping a food to keep it hot) is forbidden [even] in something that does not add heat. This is a decree lest one heat the food before wrapping it.

36b (Mishnah): If a Kirah (oven with room to put two pots on top) was heated with straw or stubble, one may put a Tavshil (cooked food) on it.

If it was heated with Gefes (the waste of olives after the oil was extracted) or wood, one may not put in it unless it was Garuf v'Katum (the coals were swept out, or ashes were put on them).

Beis Shamai permit removing from the Kirah, but not Chazarah (returning to it);

Beis Hillel permit both of these.

38a (Mishnah - Beis Hillel): One may return even a Tavshil.

(Rav Sheshes): The opinion that permits Chazarah, permits even on Shabbos.

Also R. Oshaya permits Chazarah on Shabbos;

R. Oshaya: I was once in front of R. Chiya in the upper story. They brought a kettle of hot water from the ground floor, mixed a cup of wine for him, and returned the kettle [to the Kirah]. He did not comment.

(Rav Tadai): One may return only if the pot is still in his hand, but not if he put it on the ground.

(R. Ami): R. Chiya disagrees, and permits even if he put it on the ground.

(Rav Dimi or Rav Shmuel bar Yehudah): R. Elazar permits only if the pot is still in his hand, but not if he put it on the ground.

(The other of Rav Dimi and Rav Shmuel bar Yehudah): R. Elazar permits even if he put it on the ground.

Version #1 (Chizkiyah citing Abaye): The Heter to return if it is still in his hand is only if he removed it with intent to return it.

Inference: If he put it on the ground it is forbidden even if he intended to return it.

Version #2 (Chizkiyah citing Abaye): The Isur to return if put it on the ground is only if he removed it without intent to return it.

Inference: If it is still in his hand, he may return it even if he removed it without intent to return it. (end of Version #2)

Questions (R. Yirmeyah): If he hung it on a stick, or put it on a bed, what is the law?

Question (Rav Ashi): If he transferred the water from one pot to another, what is the law?

These questions are not resolved.

Rishonim

Rambam (Hilchos Shabbos 3:10): Anything that one may be Meshaheh on a fire, when one removes it on Shabbos, he may not return it to its place. We return only on a Kirah Garuf or Katum, or a Kirah or Kupach (an oven with room for one pot on top) heated with straw or stubble. This is only if he did not put the pot on the ground. Once he put it on the ground, he may not return it, even on a Kirah Garuf or Katum. One may not return to a Tanur (an oven that retains heat very well) or Kupach heated with Gefes or wood, even if it was Garuf or Katum, for they are very hot.

Ra'avad: It seems that the text of the Rambam should say "whatever is forbidden to be Meshaheh..."

Rosh (3:2): Rashi explains that Rav Sheshes teaches that Chazarah is permitted not only at night, rather, even on Shabbos day. We do not say that surely he removed it without intent to return it, and now it is a new Shehiyah. The Gemara connotes unlike this. "Shabbos" includes the night. Rather, "even on Shabbos" teaches that the argument is primarily before Shabbos. This is why it says "until he sweeps or puts ashes". Had it discussed only on Shabbos, it would have said "unless he swept or put ashes."

Question: If it is forbidden during the day, what is the limit? Will we say that one who removed the pot in the middle of the day may not return it?!

Answer (Rosh): One may not return it if it cooled off, and there is no time to boil it during the day. However, if so we must say that Shabbos is unlike Erev Shabbos, and on Shabbos the stringent opinion forbids Chazarah even if it is boiling. If not, what is the source that R. Oshaya agrees that one may return [on Shabbos]? Perhaps the water was still boiling!

Note: Perhaps the Gemara records that he was in the upper story and the water was brought from and returned to the ground floor, to teach that there was a time delay and surely it ceased boiling!

Rosh (ibid.): "Even on Shabbos" connotes that the argument is the same on Erev Shabbos and Shabbos. Also on Erev Shabbos, even if it is boiling it is forbidden, if it is so close to dark that if it were cold, there would not be time to boil it before Shabbos. Beis Shamai forbids Chazarah during the day even if it is Garuf v'Katum, similar to on Shabbos. It is all one decree. Since it does not cook any more on the Kirah, this is like a new Shehiyah. If we permit Chazarah during the day, people will permit on Shabbos. As long as it was not fully cooked, one may return the entire day.

Rosh (2): The questions about hanging the pot on a stick, putting it on a bed and emptying a kettle into another were not settled. We are stringent.

Rosh (4:2): The custom is to sweep [out the coals] and remove the pot, without concern for where they put it [and to return the pot afterwards]. According to one version of Chizkiyah, if one placed the pot on the ground, he may not return it even before Shabbos, like I explained above (4:1)! Perhaps the custom is like the other version, which says that intent to return helps even if he put it on the ground. If it is still in his hand, this helps even if he did not intend to return it. We are lenient about mid'Rabanan laws. R. Chananel says that due to the stringency of Shabbos, this is like a Torah law, and we are stringent. I say that we are stringent about Chazarah before Shabbos only when he took from the Kirah on which he will do Shehiyah. If we permit Chazarah during the day, people will do Chazarah after dark. If he returned it for the sake of Hatmanah, it is permitted.

Poskim

Shulchan Aruch (OC 253:2): If a Kirah was Garuf v'Katum, and one took a pot from it even on Shabbos, he may return it as long as it is boiling (Rema - and still in his hand) and he did not put it on the ground.

Magen Avraham (20): The Beis Yosef forbids only if he put it on the ground. If he put it on a bed or bench, or poured from one kettle to another, it is permitted. However, the Rema forbids all of these. It is like putting [anew] on the fire on Shabbos.

Gra (DH v'Odo): The Rema is like the Rosh, who says that we are stringent about the questions.

Gra (DH v'Lo): The Rambam is lenient about the question, like the Yerushalmi. He always does so (follows the Yerushalmi when the Bavli did not decide - Damesek Eliezer). The Yerushalmi says that if he hung it on a peg and it is still warm he may return it. However, afterwards it says that if it left his hand it is forbidden, like the Rema.

Mishnah Berurah (52): If it is not Garuf v'Katum, or in a Tanur, it is forbidden even if it is Mitztamek v'Ra Lo.

Mishnah Berurah (53): One may return even on Shabbos, even many times.

Mishnah Berurah (55): If one put it on a bed, bench, etc., the initial Shehiyah is Batel. Chazarah is like a new Shehiyah. It looks like he cooks on Shabbos.

Kaf ha'Chayim (38,39): Many Yir'ei Shamayim and people meticulous about their deeds do not do Chazarah at all. Such a person should not be stringent when he eats with others. Binos Deshe says that this is one of the reasons not to eat with others on Shabbos. However, if one must do so, he must eat from the Tavshil like everyone else, to avoid haughtiness and "Lo Sisgodedu."

Kaf ha'Chayim (40): Sefer ha'Terumah, the Mordechai and Ri'az are lenient even if he put it on the ground, as long as he intended to return it.

Rema: And he intended to return it.

Magen Avraham (21): The Beis Yosef permits even if he did not intend to return, like the Rambam wrote.

Gra (DH v'Daito): This is stringent like the first version, like the Rosh brings from R. Chananel. Due to the stringency of Shabbos, we are stringent like for Torah laws. The Rambam and Shulchan Aruch omitted Abaye, like the Yerushalmi, which omitted him.

Mishnah Berurah (56): Many Rishonim are lenient if it is still in his hand even though he did not intent to return it, and if he intended to return it, even if he removed it from his hand. When needed, one may rely on this.

Rema (ibid.): This is only if it was fully cooked.

Taz (10): The Rema (Teshuvah 102) permits to open the oven on Shabbos night, and a Nochri will return the pot and seal the oven, if it was cooked k'Ma'achal Ben Drusai.

Kaf ha'Chayim (42): If he intended to return it after it cooled off, it is forbidden even while it is still boiling.

Rema (ibid.): Some say that the Isur is only if he removed from the oven during the day and did not return until dark, but if he removed after dark, he may return even if he put the pot on the ground.

Beis Yosef (DH u'Mah she'Chasav Rabeinu Kol): R. Yerucham wrote in the name of R. Yonah that one may return it only if it reached k'Ma'achal Ben Drusai before Shabbos. Also the Magid Mishneh said so in the name of the Ramban and Rashba, but he did not mention 'before Shabbos.' This seems correct. If it is cooked now that he removes it from the fire, what difference does it make whether or not it was cooked before Shabbos? Perhaps R. Yonah discusses one who removed it just when it became dark. If it was not k'Ma'achal Ben Drusai before Shabbos, also now it is not. The Tur wrote that if it was not fully cooked, even if it was k'Ma'achal Ben Drusai, the Melachah of cooking applies, even if it is boiling. If so, the Heter of Chazarah here is if it was fully cooked. If not, one may not return it as long as the Kirah is still hot.

Mishnah Berurah (65): This is even if he did not intend to return it, since all know that it was on the Kirah and it already cooked.

Rema (ibid.): The custom is to be lenient with our ovens, which are considered like Kiros. It is good to be stringent. If it cooled off, all forbid.

Taz (12): The Ran brings this leniency, but Tosfos ha'Rosh and the Rosh and those drawn after them disagree.

Kaf ha'Chayim (69): The custom is to be lenient, like the Rema says.