Gemara

(Rav): Me'ilah does not apply to ashes of the Tapu'ach (the heap on top of the Mizbe'ach);

(R. Yochanan): Me'ilah applies to them.

They argue about after Terumas ha'Deshen. Rav says that Me'ilah does not apply, for the Mitzvah has been finished;

R. Yochanan holds that since "v'Lavash ha'Kohen" obligates wearing Bigdei Kehunah, this shows that the ashes retain their Kedushah, and Me'ilah still applies.

Pesachim 21a (Mishnah): After the permitted time for Chametz on Erev Pesach expires, it is Asur b'Hana'ah. One may not use (burn) it to heat an oven;

R. Yehudah says, Bi'ur is fulfilled only through burning;

Chachamim say, one may even crumble it and cast it into the wind or the sea.

21b: 'One may benefit from it' discusses Chametz that was charred before (the time of Isur). It permits benefit even after Chametz is forbidden;

'One may not use it to heat an oven' is a Chidush according to R. Yehudah, who holds that Bi'ur is only through burning. One might have thought that one may benefit from it while burning it. The Mishnah teaches that this is not so.

27b (Beraisa - R. Yehudah): A Kal va'Chomer mandates burning to fulfill Bi'ur. Bal Yera'eh and Bal Yimatzei do not apply to Nosar, yet Nosar must be burned. They apply to Chametz, all the more so it must be burned!

Chachamim: You cannot make a Kal va'Chomer if the stringency leads to a leniency!

If someone has no wood (to burn Chametz), you exempt him from destroying it. "Tashbisu Se'or mi'Bateichem" - obligates destroying it any way you can!

Temurah 33b (Mishnah): The following are Nikbarim (must be buried) -- an ox sentenced to be stoned...

The following are Nisrafim (must be burned) -- Chametz during Pesach...

One may not burn Nikbarim. One may not bury Nisrafim;

R. Yehudah permits to be stringent and burn Nikbarim;

Chachamim forbid, because ashes of Nisrafim are permitted, but ashes of Nikbarim are forbidden. (If Nikbarim are burned, people might benefit from the ashes.)

Avodah Zarah 43b (Mishnah - R. Yosi): One grinds up idols and throws the dust to the wind, or casts it to the sea.

Chachamim: No, this will fertilize the ground on which it lands, and we may not derive any benefit from idols - "Lo Yidbak b'Yadcha Me'umah Min ha'Cherem"!

Rishonim

Rambam (Hilchos Chametz 3:11): Bi'ur Chametz is through burning, or crumbling it and casting it into the wind or the sea. If one burned Chametz before the sixth hour, one may benefit from the coals during Pesach. If he burned it after the sixth hour, since it is Asur b'Hana'ah, one may not use it to heat an oven or cook. If he used it to cook or bake, the bread or cooked food are Asur b'Hana'ah. Likewise, the coals are Asur b'Hana'ah, since he burned it after it was Asur b'Hana'ah.

Magid Mishneh: The coals are forbidden, for this is like charring it after the time. Chametz is not Nisrafim, therefore the ashes are forbidden.

Rambam (Hilchos Pesulei ha'Mukdashim 19:14): One may not bury Nisrafim, and one may not burn Nikbarim. If one is stringent to burn them, he is lenient about the ashes, for the ashes of Nikbarim are forbidden.

Tosfos (Temurah 33b DH ha'Nisrafim): Since the Torah commanded to burn Nisrafim, after one did so, it is as if the Mitzvah was done. Me'ilah never applies after the Mitzvah was done. Here also, after the Mitzvah was done, the Isur vanishes. Even though the Torah commanded to burn an Asherah, it is always forbidden, for it says "Lo Yidbak..." The Torah did not command to burn Nikbarim, so their Isur is permanent.

Pnei Yehoshua (Pesachim 27b DH Chametz): Even though we hold that the ashes of Nikbarim are forbidden, for Chametz this is only mid'Rabanan. The Torah forbids Isurei Hana'ah only in the normal way to benefit from them. Surely, ashes of Chametz is not the normal Hana'ah! Further, I say that the Isur of the ashes of all Nikbarim is only mid'Rabanan, unlike Tosfos. This can resolve why the Rambam rules like Chachamim regarding Chametz, yet permits burning it.

Rosh (Pesachim 2:2, citing R. Yonah): We hold that Chametz need not be burned, therefore the ashes are forbidden, and all the more so the coals.

Tosfos (26b DH Bishlah): Even though the ashes of Nisrafim are permitted l'Chatchilah, coals are different.

Poskim

Shulchan Aruch (OC 445:1): Bi'ur Chametz is through burning, crumbling and throwing to the wind, or throwing to the sea.

Magen Avraham (1): Our texts of the Gemara say that Chachamim permit even to crumble it. The texts of the Rif and Rosh connote they permit only crumbling it. The Tur questioned why the Rambam permits burning it. Perhaps the Gemara forbids burning it like Nisrafim, i.e. to benefit from the ashes. We do not decree lest he benefit from the ashes.

Chak Yakov (1): We must say that the Rambam holds that one may not burn Nikbarim only regarding those listed in the Mishnah. This excludes Chametz, for Chachamim explicitly taught that it may be burned or buried. It is not among Stam Nikbarim. Therefore, the Rambam did not list it among Nisrafim and not among Nikbarim. Emunas Shmuel (55) says so. Do not ask that people will come to be lenient about the coals and ashes. Chametz is more stringent than other Isurei Hana'ah, i.e. there is Bal Yera'eh and Bal Yimatzei. It is better to be stringent to burn it, lest people transgress Bal Yera'eh and Bal Yimatzei, even though this is a leniency regarding the ashes. Also, since everyone eradicates it properly before six hours, the coals and ashes are permitted b'Hana'ah. Only if one did not do Bi'ur (in the proper time), he does Bi'ur afterwards (and then the ashes are forbidden). This is uncommon; Chachamim did not decree about it. I say so according to the Rambam. What the Magen Avraham wrote is difficult.

Shulchan Aruch (2): If one burned Chametz before the sixth hour, one may benefit from the coals during Pesach. If he burned it after the sixth hour, since it is Asur b'Hana'ah, one may not use it to heat an oven or cook. If he used it to cook or bake, the bread or cooked food are Asur b'Hana'ah. Likewise, the coals are Asur b'Hana'ah, since he burned it after it was Asur b'Hana'ah.

Beis Yosef (DH Aval): Benefit from the ashes after burning it depends on the argument of R. Yehudah and Rabanan. Whatever must be burned, the ashes are permitted.

Bach (2): R. Yehudah admits that one may not benefit from it while burning it if it is already forbidden, like the Gemara says. The custom is to be stringent to burn it like R. Yehudah, and to forbid Hana'ah from the ashes, like Chachamim.

Avnei Milu'im (Teshuvah 19): The Rambam rules like Chachamim, that one may not burn Nikbarim, for this stringency leads to a leniency, i.e. benefit from the ashes. The Beis Hillel (YD 294:1) asked why the Rambam permits burning Chametz. The ashes and coals are forbidden! His text said that Chachamim say that even crumbling (and casting to the wind...) is Bi'ur. Since burning also fulfills the Mitzvah, the ashes are permitted. Even according to Tosfos and the Rosh, whose text says 'Chachamim say that crumbling it is Bi'ur', the Mitzvah is fulfilled through burning. It is no worse than crumbling. R. Yehudah holds that the Mitzvah is only burning. He learns a Kal va'Chomer from Nosar. Chachamim disqualify a Kal va'Chomer if its stringency leads to a leniency, to be exempt if he cannot burn it. This shows that burning is included in Bi'ur. Once the Mitzvah was done, there is no Me'ilah, therefore the ashes are permitted. We do not say so for other Nikbarim, for there is no Mitzvah to remove them from the world. We merely cover them to hide them, lest one stumble through benefit from them. Hagahos Maimoniyos (Hilchos Chametz 3:20) forbids to cast Chametz in a Hefker place, for the Mishnah obligates burial. The Yerushalmi forbids giving even to a dog that is not his. This shows that there is no Mitzvah to bury Nikbarim. We must merely hide it, so no living being will eat it. When there is a Mitzvah to destroy it, after burning the ashes are permitted.

Avnei Milu'im (ibid.): R. Yosi permits to throw the dust of idols to the wind. Chachamim forbid, for this will fertilize the ground, and it says "Lo Yidbak..." Chachamim permit to grind up Chametz and throw it to the wind, because its Mitzvah was done. Ashes of Nikbarim are forbidden only for Isurei Hana'ah for which there is no Mitzvah to destroy them, only to hide them, and this was not fulfilled. "Lo Yidbak..." forbids idolatry even after the Mitzvah was fulfilled. The Gemara suggested that perhaps R. Yehudah would permit benefit from the fire while burning Chametz. Since Chachamim agree that one fulfills the Mitzvah through burning, the Hava Amina should apply also to them! The Heter of ashes of Nisrafim is only after they were burned. While burning it is forbidden, except for Tamei Terumah, for which a verse permits benefit while burning it. The Hava Amina was that R. Yehudah equates Chametz to Tamei Terumah. There was no such Hava Amina for Chachamim, for they do require burning Chametz.

Avnei Milu'im: The Rambam teaches that the coals are permitted only if he burned it before six hours. He did not teach that the ashes are forbidden. It seems that he permits them. Tosfos distinguishes between ashes and coals. The Rambam forbids coals. It seems that this is a decree lest one benefit from the flame (while burning the Chametz). The Magid Mishneh says that the coals are forbidden, like Chametz charred after the time, and ashes of Chametz are forbidden. I say that the ashes are permitted. This answers Tosfos' question, why the Mishnah did not list Chametz among Nisrafim or Nikbarim, for even Chachamim, who says that it is Nikbarim, permit the ashes because the Mitzvah was done.

Kol Mevaser (1:16): The Mishnah explicitly lists Chametz among Nisrafim, and the Gemara deduces that the Mishnah is R. Yehudah! Perhaps the Avnei Milu'im saw Chak Yakov's question why the Rambam did not list it among either, and erred about this. (Perhaps the question was why the Mishneh Torah did not list it - PF.)

Mishnah Berurah (12): The coals and ashes of Chametz are forbidden. Some Acharonim forbid what was cooked in an oven heated through the ashes and coals, even if they were swept out before the food was put in.

Kaf ha'Chayim (20): If the outside was charred, but inside it is edible, it is Asur b'Hana'ah.

Mishnah Berurah (14): One may not benefit from the warmth or light of the coals. One may use the ashes to cover the blood of a slaughtered bird or Chayah, for Mitzvos Lav Leihanos Nitnu (regarding Isurei Hana'ah, fulfillment of Mitzvos is not called Hana'ah).

Shulchan Aruch (EH 28:21): If one was Mekadesh with Chametz when it is forbidden mid'Oraisa, she is not Mekudeshes.

Bach (DH Kosvu): One answer in Tosfos (Pesachim 29b DH Ein) is concerned for Kidushin if the ashes of the Chametz are worth a Perutah, for we hold that Chametz must be burned, so the ashes are permitted.

Chelkas Mechokek (39): The Shulchan Aruch (OC 445:1) says that Chametz can be destroyed in any way, i.e. to be stringent (if one cannot burn it). Perhaps for Kidushin, we are concerned lest it must be burned. However, if she does not know that it is Asur b'Hana'ah, surely she is not Mekudeshes.