Gemara

(Mishnah): If one cleared a kettle, one may not put water in it to get hot, but one may put water in it or in a cup to become lukewarm.

41b (Abaye): If one emptied the water out of a kettle, one may not put water in it at all, for it is Metzaref;

The Mishnah is like R. Yehudah, who forbids Davar she'Eino Miskaven.

Shmuel: It is permitted even to be Metzaref.

Objection: Surely, one may not intend to be Metzaref!

Correction: Rather, Shmuel permits even a Shi'ur to be Metzaref.

Inference: Shmuel holds like R. Shimon (who permits Davar she'Eino Miskaven).

Question: Shmuel permits to extinguish (on Shabbos) a metal coal in Reshus ha'Rabim to save people from being harmed, but not a coal of wood;

If he held like R. Shimon, [to prevent harm to the Rabim] he should permit extinguishing even a wooden coal! (This is Melachah she'Einah Tzerichah l'Gufah. It is not mid'Oraisa!)

Answer: He holds like R. Shimon regarding Davar she'Eino Miskaven, but he is Mechayev for Melachah she'Einah Tzerichah l'Gufah like R. Yehudah.

134a (Beraisa #1): One may not sweeten mustard through extinguishing a coal in it.

Contradiction (Beraisa #2): One may sweeten it with a coal.

Answer: It is permitted with a metal coal, but forbidden with a coal from wood.

Question (Abaye): Why is this different than roasting meat over coals?

Answer (Rav Yosef): There, it is impossible. Here, it is possible.

Rishonim

Rif and Rosh (Beitzah 12a and 2:23): One may sweeten mustard with a metal coal, but not with a coal from wood. This different than roasting meat over coals. There, there is no Kivuy (extinguishing). Here, there is Kivuy.

Rebuttal (Ba'al ha'Ma'or DH v'Ha): Our text is unlike this, for any Kivuy for the sake of food is permitted. Why should it be unlike burning?! Rather, the text says 'there, it is impossible. Here, it is possible.' I.e. one cannot eat roasted meat without roasting. It is (not - the Bach deletes this) possible to use mustard without sweetening it. It is needed only for the most finicky, pampered people. It is not something that everyone does. Melachah on Yom Tov is not permitted in such cases.

Defense (Milchamos Hash-m): Meleches Ochel Nefesh is permitted for everyone on Yom Tov. Anyone may slaughter a deer, even though only rich people eat deer meat. What will one do if he has no other accompaniment? Everyone sweetens mustard. The Rif's text is correct. Sweetening mustard is a Tikun (fixing). It is not included in cooking and baking. Rather, it is like grinding. Yalkut Shimoni says 'everything created in the six days must be fixed - mustard must be sweetened, wheat must be ground...' The Hava Amina was that it is forbidden because the Kivuy is not for the sake of Ochel Nefesh. We conclude that it is Kivuy, but there is no Kivuy when meat roasts on coals.

Rambam (Hilchos Shabbos 12:2): One is liable for extinguishing any amount, whether he extinguishes a Ner or a coal from wood. However, one who extinguishes a metal coal is exempt. If he intended to be Metzaref he is liable, for this is the Melachah of iron forgers. They heat up iron until it glows like a coal, then they extinguish it in water to forge it. One is liable for this Tziruf for it is a Toldah of Kivuy. One may extinguish a metal coal in Reshus ha'Rabim to prevent harm to the Rabim.

Rebuttal (Ra'avad): Putting a metal coal in water is not Kivuy (mid'Oraisa), but it is Makeh b'Patish, for this completes the forging. In any case he is exempt because Tziruf is mid'Rabanan.

Question (Tosfos 41b DH Meicham): Rashi (Yoma 34b DH b'Chol) says that R. Yehudah permits Davar she'Eino Miskaven regarding Tziruf, for it is only mid'Rabanan. (The text there should not say 'because Ein Shevus b'Mikdash'.) If so, why does R. Yehudah forbid extinguishing a metal coal to avoid harm to the Rabim? And why does he forbid dragging a bed on a dirt floor? Even if he makes a furrow, this is ki'Le'acher Yad (unskillfully), which is only mid'Rabanan!

Answer #1 (Tosfos ibid.): Even though Tziruf is mid'Oraisa, the Gemara in Yoma calls it mid'Rabanan, and permits because Ein Shevus b'Mikdash, because R. Yehudah forbids Davar she'Eino Miskaven only mid'Rabanan.

Answer #2 (Magid Mishnah): In Yoma, Abaye says that Tziruf is only mid'Rabanan. The Halachah follows Rav Bivi, who says that it is mid'Oraisa.

Answer #3 (Ramban 42a DH Mechavin): Tziruf is mid'Oraisa only if one intends to make a Keli.

Answer #4 (R. Chananel 42a DH Lemeimra, and BaHaG): R. Yehudah always forbids Melachah she'Einah Tzerichah l'Gufah. Shmuel permits it only to prevent damage to the Rabim (perhaps it is considered Piku'ach Nefesh), like he permits trapping snakes. A metal coal is dangerous even when it ceases to be red, so people do not know to beware. A wooden coal is dangerous only while it is glowing, and people know to avoid it then.

Ramban (ibid.): BaHaG and R. Chananel hold that Tziruf is always mid'Oraisa. In Yoma it is called mid'Rabanan because the iron made the water warm. Tziruf requires cold water.

Answer #5 (Shabbos 42a DH Gacheles): All Tziruf is mid'Rabanan. Even R. Yehudah permits in order to avoid harm to the Rabim.

Rebuttal (of Rambam - Ra'avad): Since the Rambam says that Kivuy applies to a metal coal, he should be Mechayev regardless of the intent, for it is a Pesik Reishei. The Rambam is Mechayev for Melachah she'Einah Tzerichah l'Gufah!

Defense (Magid Mishnah): If one does not intend, we cannot say that he did a Melachah. One who breaks a chip of wood to make a toothpick is liable. One who did so without this intent is exempt [even according to R. Yehudah]!

Lechem Mishneh (1): This applies only to extinguishing a metal coal. This was not done in the Mishkan. The Rambam is Mechayev for Tziruf only when one intends to make a Keli, for iron forgers do so. If he did not intend for this, he did not accomplish anything. However, one who is Metzaref a kettle or any other Kli strengthened it. R. Yehudah is Mechayev regarding of intent. The Gemara holds that R. Yehudah would forbid putting thick pieces of iron in the water, because they are Kelim.

Question (Ra'avad): The Rambam says that burning and extinguishing a metal coal is mid'Oraisa, but Shabbos 134a permits doing so in mustard!

Note: The Rambam said only that burning it is mid'Oraisa. He says that Kivuy is mid'Rabanan, but one who intends for Tziruf is Chayav.

Answer (Ramban ibid.): Tziruf is only in water.

Rebuttal (of Rambam - Ramach, cited in Kesef Mishneh): The Gemara says that Kivuy of metal is Asur. It does not say that it is mid'Oraisa!

Rambam (Hilchos Yom Tov 4:6): One may sweeten mustard with a metal coal, but not with a coal from wood.

Poskim

Shulchan Aruch (OC 334:27): One may extinguish a coal in a place where many walk, to prevent damage to the Rabim, whether it is of metal or wood.

Mishnah Berurah (82): If one can move it to where it will not damage, this is better than extinguishing it.

Mishnah Berurah (83): This is not Metzaref, because he does not intend for it.

Mishnah Berurah (84): We exempt Melachah she'Einah Tzerichah l'Gufah, so one is liable for Kivuy only if he intends to make a wick. Nevertheless, one may not extinguish to avoid monetary loss.

Shulchan Aruch (ibid.): The Rambam forbids extinguishing a wooden coal.

Kaf ha'Chayim (157): The Halachah is unlike the Rambam. Nevertheless, the Shulchan Aruch brings his opinion to show that l'Chatchilah one should be stringent.

Shulchan Aruch (510:3): (On Yom Tov) one may not sweeten mustard through extinguishing a wooden coal in it. It is permitted with a metal coal.

Beis Yosef (DH Ein): It is forbidden with a wooden coal because sometimes it is a Melachah to extinguish it (if he intends to make a wick), and he could have sweetened the mustard from before Yom Tov.

Taz (5): It is forbidden with a wooden coal due to Kivuy. Kivuy does not apply a metal coal, for it does not become coal. I do not understand Milchamos Hash-m. If sweetening mustard is like grinding, also baking is Tikun, even more than grinding! Really, grinding is not forbidden due to Tikun, rather, like I explain in 495:2 [because one often does much at a time. If it were permitted, people would neglect Simchas Yom Tov.] The Ba'al ha'Ma'or's text is correct. Here, sweetening mustard, is possible before Yom Tov. Roasting meat cannot be done before Yom Tov, for the taste would be blunted, like the Beis Yosef says in Siman 511.

Sha'ar ha'Tziyon (511:28): There is no problem of Kivuy with incense on metal coals, like we say regarding mustard. However, the Ro'oh says that Rav Yehudah is lenient only regarding smoking Peros, but not for scent. This requires investigation.