Gemara

(Mishnah): If a Kirah (oven) 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 returning to it;

Beis Hillel permit both of these.

Question: What is the meaning of 'one may not put'?

Does it forbid Chazarah (returning to the Kirah), but Shehiyah (leaving in it when Shabbos begins) is permitted, even if it was not Garuf v'Katum? This would be like Chananyah;

(Beraisa - Chananya): Shehiyah of anything k'Ma'achal Ben Drusai (a third or half cooked) is permitted, even if the stove was not Garuf or Katum.

Or, the Mishnah forbids Shehiyah, and all the more so Chazarah. Shehiyah is permitted only if it was Garuf v'Katum. If not, it is forbidden, and all the more so returning is forbidden.

Rishonim

Rif (16a): The Gemara concludes that the Mishnah forbids Shehiyah. It is permitted only if the oven is Garuf v'Katum, and all the more so Chazarah is forbidden. Presumably, this is the conclusion, for after the discussion [to try to answer the question], the Gemara asked whether one may be Somech to a Kirah. Perhaps we are more than lenient than regarding Shehiyah in or on a Kirah, which is forbidden, or perhaps they are the same. This shows that Shehiyah is forbidden if it is not Garuf v'Katum. Also, to answer the question we brought a Beraisa that permits Semichah if it was heated with Gefes or wood, but forbids Shehiyah unless it was Garuf v'Katum. If the coals dimmed, or if one put thin chaff of flax on top of them, this is like covering with ashes. R. Oshaya and Rabah bar bar Chanah in the name of R. Yochanan forbid Shehiyah. Also, regarding an oven heated with straw or stubble, Rav Yosef wanted to permit Semichah. Abaye challenged him from a Mishnah. The Gemara said that if it is not Garuf v'Katum, surely Shehiyah is forbidden. This shows that this is the Halachah. Rav Sheshes said in the name of R. Yochanan that the Mishnah discusses Chazarah, but Shehiyah is permitted, even if it is not Garuf v'Katum. The Halachah does not follow him. Even though Rava supported him, it is not a real support. We rely on the Gemara's conclusion, that if it is not Garuf v'Katum, Shehiyah is forbidden if the water or food was not fully cooked. If it was fully cooked, even if it is not Garuf v'Katum, Shehiyah is permitted if it is Mitztamek v'Ra Lo, e.g. turnips, porridge, dates and similar things. If it is Mitztamek v'Yafeh Lo, e.g. cabbage, beans and diced meat, it is forbidden. One may be Meshaheh anything in a Kirah that is Garuf v'Katum. The decree was lest one come to stoke coals. Since it is Garuf v'Katum, he will not come to stoke. This is the Halachah.

Rosh (3:1): Rashi says that we rely on Chananyah to do Shehiyah on a Kirah that is not Garuf v'Katum. The Stam Mishnah is like him. The Gemara permits bread that formed a crust, even if it is not fully cooked. Rav Sheshes in the name of R. Yochanan explained the Mishnah to discuss Chazarah, but Shehiyah does not require Garuf v'Katum. The Amora'im who forbid hold that the Mishnah discusses Shehiyah. The Ri rules like this. He brought a proof from Rava, who is Basra, and supported Rav Sheshes. The Ba'al ha'Ma'or says so, and refuted all the Rif's proofs. There is no proof from the question about Semichah. Semichah applies to Chazarah, just like to Shehiyah. The Gemara said that if it is not Garuf v'Katum, one may not do Chazarah on or in it, but perhaps one may do Chazarah near it. A Beraisa teaches that if a Kirah was heated with Gefes or wood, one may be Somech, but he may not be Meshaheh on it unless it was Garuf v'Katum. This is like R. Yehudah. Since he permits Semichah of Shehiyah, presumably the Tana of our Mishnah permits Semichah of Chazarah. Tana'im argue about the laws of R. Oshaya and R. Chiya. Even if they hold like R. Yehudah, the Stam Mishnah is better than all of these. The Sugya in which Rav Yosef asked about an oven heated with straw or stubble discusses Chazarah. Shehiyah is permitted in every case, and all the more so Semichah. We hold like Chananyah. There are many opinions about how to rule. Yisre'elim cling to Oneg Shabbos, and they would not heed us to be stringent, so we leave everyone according to the custom based on those who rule like Chananyah.

Rambam (Hilchos Shabbos 3:4): If a Tavshil or water was not fully cooked, or the Tavshil was fully cooked but it is Mitztamek v'Yafeh Lo, one may not do Shehiyah on the fire, even if it was placed before Shabbos. This is a decree lest he stoke the coals to complete the cooking or dry it out more. Therefore, if he swept the fire or covered the fire of the Kirah with ashes or fine stubble of flax, or the coals dimmed, it is as if they are covered with ashes. Or, if he heated the oven with straw, stubble or fine dung, there are no burning coals, so one may do Shehiyah. He took his mind off the Tavshil, and we do not decree lest he stoke the fire.

Poskim

Shulchan Aruch (OC 253:1): If a Kirah is made like a pot, and one places a pot on top of it, and there is room for two pots on top of it, if it was heated with Gefes (the waste of olives) or wood, one may not be Meshaheh a Tavshil on it unless it was fully cooked, and it is Mitztamek v'Ra Lo (he is displeased the more it cooks and dries out), or if the oven was Garuf or Katum. If it was heated with straw or stubble, one may be Meshaheh on it even if it is not Garuf and not Katum.

Bi'ur Halachah (DH Tavshil): The same applies to water that was not heated as much as needed, according to this opinion, which rules unlike Chananyah.

Gra (DH Lehashhoso): This is like Rav Nachman. He is brought at the end of the Sugya. This connotes that this is the conclusion.

Shulchan Aruch (ibid.): Some say that anything cooked k'Ma'achal Ben Drusai, or it is Mitztamek v'Yafeh Lo, one may leave it on a Kirah.

Beis Yosef (DH Kirah and DH v'Chasav): The Gemara did not explicitly answer the question of whether the Mishnah discusses Chazarah or Shehiyah. The Ran and Rosh brought that the Ba'al ha'Ma'or refuted the Rif's proofs. However, R. Yonah answered for the Rif. The She'altos and Ri Bartzeloni hold like the Rif. The Rambam agrees, and the Magid Mishneh says that the Ramban agrees. Rashi, the Ran citing Rav Hai Gaon, Tosfos in the name of R. Chananel, the Rashba, Semag, Semak, Sefer ha'Terumah, Hagahos Maimoniyos, Mordechai in the name of Or Zaru'a hold that the Mishnah discusses Chazarah.

Kaf ha'Chayim (26): Since it can be eaten, we are not concerned lest he stoke without need. We require Garuf v'Katum only if it was not k'Ma'achal Ben Drusai, or for Chazarah on Shabbos, for then there is more concern for stoking.

Rema: The custom is to be lenient like the latter opinion.

Kaf ha'Chayim (23): The custom is to be lenient like the latter opinion mentioned below in the Mechaber.

Bi'ur Halachah (DH v'Nahagu): The Beis Yosef concluded with the Rosh's words that Yisre'elim cling to Oneg Shabbos, and they would not heed us to be stringent, so we leave them [to be lenient]. This connotes that we do not protest only because they would not listen. It seems that the Beis Yosef agrees. Therefore, he wrote the lenient opinion in the name of "some say." If so, l'Chatchilah one should be stringent to ensure that the food is fully cooked before dark and remove it from the fire.

Chazon Ish (Mo'ed 37:3 DH b'Mishnah): The Bi'ur Halachah means that one should remove it from the fire if it is Mitztamek v'Yafeh Lo. He need not do so if it is Mitztamek v'Ra Lo. I say that the Rosh does not mean that it is better that Yisrael be Shogeg than Mezid. Letter of the law, one who has a tradition to be lenient may be lenient, for this is a mid'Rabanan law. Gedolei ha'Olam such as the Rif and She'altos were stringent, so it would have been proper to be stringent, but often this would cause Bitul Oneg Shabbos. We find that Ge'onim were not stringent, e.g. Rav Nachman bar Yitzchak, even though he was meticulous about his deeds.

Bi'ur Halachah (ibid.): However, if he delayed, e.g. guests came [shortly] before Shki'ah and he needed to cook a Tavshil for them, he may put it on a tripod even if it will cook only half-way before Shki'ah. He may remove from it at night [without concern lest this cause the coals to extinguish], for it is not resting on the coals.