Gemara

18b (Beraisa): A baker may not fill a flask with water and put it in an oven just before Shabbos.

Rejection: This is even like Beis Hillel. This is a decree, lest one stoke the coals [on Shabbos, to hasten the cooking].

Question: If so, they should forbid leaving dye in a pot!

Answer (Shmuel): They permit when the pot is off the fire.

Question: Still, it should be forbidden, lest he stir the pot!

Answer: The case is, the pot is plastered shut.

34a (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.

38b (Rav Yehudah bar Shmuel): At first, if one cooked on Shabbos, he was permitted to eat it if he was Shogeg, but not he was Mezid. The same [distinction between Shogeg and Mezid] applied to one who forgot [to remove the pot, and was Meshaheh];

Many people did Shehiyah b'Mezid and said that they forgot. Chachamim decreed to forbid even one who forgot.

39a (Mishnah): Any food that that was soaked in hot water before Shabbos, one may soak it in hot water on Shabbos.

40b (Beraisa): [On Shabbos] one may leave a flask of water in front of a fire, not to get hot, rather, only to blunt its chill.

Rishonim

Rambam (Hilchos Shabbos 9:3): If one cooks on the fire something fully cooked or something that does not need to cook at all, he is exempt.

Rosh (3:10): The Beraisa permits putting water near a fire, if there is no concern for cooking. Why do we forbid putting on a Kirah, even if it is Garuf v'Katum, even though the Tavshil was cooked before Shabbos? Cooking does not apply (any more), like the Mishnah about a food soaked in hot water before Shabbos. I answer that putting near a fire is Semichah, which is permitted. I explained that Ein Bishul Achar Bishul. The Gemara forbids Hatmanah after dark lest one heat the food before wrapping it [even though it was already cooked]. We must say that really, the decree is lest one stoke. The Gemara mentioned heating the food, for this is what is needed for Hatmanah.

Ran (19a DH Tanu): Rashi explains that the Beraisa forbids leaving water near the fire long enough to get hot. This implies that one may put it in a place where it could cook, as long as he removes it before it cooks. He agrees that on the fire is forbidden, for it looks like cooking. One may not return to a Kirah that is not Garuf v'Katum even a Tavshil that was fully cooked and is Mitztamek v'Ra Lo (further cooking is detrimental). All the more so one may not put there initially. The Rambam forbids putting it to thaw in a place where it could get cooked. However, this is only if it was not cooked. If it was cooked, one may put it even in a Kli Rishon or near the fire where it can become Yad Soledes Bo, for this is not the way of cooking. We learn from the Mishnah of anything soaked in hot water before Shabbos. Also, one may put it on top of a pot on the fire.

Hagahos Ashri (3:2): If one returned b'Shogeg, this requires investigation whether one may eat it. Perhaps we forbid only one who forgot a pot (and did Shehiyah) and it cooked, for the food was fixed on Shabbos b'Isur, but regarding Chazarah, it was already proper to eat and he does not benefit so much. Even though when a Nochri did for a Yisrael, we permit in such a case that the Yisrael does not benefit so much, when a Yisrael himself transgressed, perhaps Chachamim made a fine.

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.

Beis Yosef (DH u'Mah she'Chasav Rabeinu Kol): If the pot was in his hand until it cooled off, surely one may not return it, for this is like cooking. The Tur wrote like this in Siman 318. Also R. Yerucham wrote so in the name of R. Yonah that if it is mostly sauce and it is Mitztamek v'Yafeh Lo (further cooking is beneficial) and it is cold, when he returns it to the Kirah and it dries out, this is absolute cooking. It is clear from the Gemara that if it did not cool off, even if it ceased boiling a little, one may return it. He also 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.

Magen Avraham (19): "Boiling" means Yad Soledes Bo, unlike Olas Shabbos. The custom is to be lenient if it did not totally cool off, like the Rema says in 318:15. See the Rema here in Sa'if 5, unlike the Bach. If it cooled off, this (Chazarah) is cooking.

Rebuttal (Eliyahu Rabah 19): Siman 318 discusses Semichah. One may not return on a Kirah even if it did not totally cool off. The Bach and Olas Shabbos discuss this.

Gra (DH Kol): Since one may return to the fire, this shows that Ein Bishul Achar Bishul. Even so, if it cooled off, Bishul applies. We forbid Hatmanah after dark, lest one heat the food before wrapping it [even though it was already cooked]. See the Rosh (3:10, who explained differently).

Mishnah Berurah (54): If it is no longer Yad Soledes Bo, the Isur Bishul applies. Based on the Rema (318:15), we are lenient as long as it did not totally cool off.

Kaf ha'Chayim (38): The Mechaber requires that it is still boiling. It is not enough that it is still hot, and there is no Bishul Achar Bishul. It is like a new Shehiyah on Shabbos. The Bach says so, and the Shulchan Aruch connotes like this. Even though the Beis Yosef brought unlike this from R. Yerucham, he did not rely on this in the Shulchan Aruch. The Beis Yosef (Siman 318) wrote that Ein Bishul Achar Bishul if it is still Yad Soledes Bo. He did not rely on this regarding Chazarah, rather, only for putting it on a wrapped pot to keep it warm. The Eliyahu Rabah rejected the Magen Avraham. Therefore, many Yir'ei Shamayim and people meticulous about their deeds do not do Chazarah at all. Firstly, perhaps it ceased boiling immediately after he removed it from the fire, and he will not notice. Also, perhaps commoners will see and learn from him to do Chazarah, but they will not know the necessary conditions.

Shulchan Aruch (1): If one forgot and was Meshaheh, if it was fully cooked, it is permitted. If it was partially cooked, it is forbidden until Motza'ei Shabbos. If he was Meshaheh b'Mezid, both of these are forbidden.

Rema: They are forbidden until bi'Chdei she'Ya'asu (the time to do so after Shabbos). If a Nochri returned it on Shabbos, it is as if he forgot and was Meshaheh.

Taz (8) and Magen Avraham (13): Even if it is Mitztamek v'Yafeh Lo, he did not benefit much, for it was already edible.

Magen Avraham (12): The Nochri returned it for the sake of the Yisrael.

Kaf ha'Chayim (24): The Levush says that if the Yisrael told the Nochri to do so, it is as if the Yisrael did so himself, and it is forbidden.

Gra (DH v'Im): Mezid through a Nochri for the sake of Yisrael is like Shogeg through Yisrael. We hold like R. Yehudah, who forbids forever if a Yisrael was Mezid, and through a Nochri he permits on Motza'ei Shabbos bi'Chdei she'Ya'asu.

Rema (ibid.): If a Yisrael returned it, it is as if he intentionally was Meshaheh. If it is Mitztamek v'Ra Lo, it is permitted, for he did not benefit from the Isur.

Beis Yosef (DH Kasuv): Hagahos Mordechai (79) says that whenever one transgressed a mid'Rabanan law, whether Shehiyah, Chazarah or Hatmanah, one may not eat it unless Chachamim explicitly permitted. R. Yehudah permits [b'Di'eved] Mitztamek v'Ra Lo. Do not say that this is only for Meshaheh, but perhaps he forbids Chazarah, for the person did an act. This is wrong. Since he does not benefit through returning it, it is permitted.

Magen Avraham (14): This is even if the Yisrael returned it b'Shogeg. The Mordechai forbids if one forgot and was Meshaheh if it is Mitztamek v'Yafeh Lo, so the Rema rules like him regarding Chazarah. I say that it is permitted to others, for Hagahos Ashri was unsure whether we forbid even to the transgressor, since he was Shogeg and it was already cooked.

Kaf ha'Chayim (25): Eliyahu Rabah says that it is permitted to others only according to the latter opinion [in the Mechaber, which permits Shehiyah of what reached k'Ma'achal Ben Drusai], but the first opinion forbids even to others.

Gra (DH v'Im): Even b'Shogeg it is forbidden if Mitztamek v'Yafeh Lo. It is as if it was not fully cooked. Even Chananyah permits only Shehiyah (of what is k'Ma'achal Ben Drusai), but not Chazarah.