OPINIONS: The Mishnah (33b) lists the various forbidden items that must be burned. Among these items is Chametz on Pesach, which must be burned and not merely buried. The Gemara explains that this Halachah follows the opinion of Rebbi Yehudah in Pesachim (21a), who says that Chametz must be destroyed only by burning and by no other means. This implies that the Mishnah does not follow the view of the Rabanan in Pesachim, who maintain that one may destroy his Chametz in any manner (even by crumbling it up and throwing it into the sea). According to the Rabanan, Chametz is in the category of things that are buried and not burned. This is stated clearly by the TUR (OC 445) and implied by TOSFOS in Pesachim (21b, DH Behadei).
The Gemara earlier states that the ashes of forbidden items that are buried remain forbidden from benefit, unlike the ashes of forbidden items that are burned. What is the Halachic status of ashes of Chametz? Since, according to Rebbi Yehudah and the Mishnah here, Chametz must be burned, is one permitted to benefit from the ashes of burned Chametz on Pesach?
(a) The ROSH in Pesachim writes that since the Mishnah here follows the view of Rebbi Yehudah, many Rishonim, including RASHI, rule like Rebbi Yehudah. Since Chametz must be burned, the ashes are permitted.
(b) The Rosh continues and quotes the GE'ONIM, BA'AL HA'ITUR, and RABEINU YONAH who rule like the Rabanan who say that Chametz may be disposed of in any manner. Accordingly, the ashes of burned Chametz remain forbidden.
This also seems to be the opinion of the RAMBAM (Hilchos Chametz u'Matzah 3:11) who writes, "How does one do Bi'ur Chametz? He burns it, or breaks it up and spreads it into the wind, or throws it into the sea." Since the Rambam says that one may destroy Chametz in ways other than burning, it is clear that he rules like the Rabanan. This is also the ruling of the SHULCHAN ARUCH (OC 445:1).
The MAGEN AVRAHAM questions this ruling. If Chametz is among the things that are buried and not burned, this means that it may not be burned, as the Mishnah explicitly states that things that must be buried must not be burned. However, the Rabanan certainly do not prohibit one from burning his Chametz!
The AVNEI MILU'IM (#19) and others explain the Rambam's ruling based on the principle expressed by Tosfos earlier (33b, DH ha'Nisrafin). Tosfos asks what the reasoning is behind the Gemara's distinction (34a) that the ashes of items that must be buried are forbidden from benefit, while the ashes of items that must be burned are not forbidden from benefit. Why are the ashes of burned items permitted, while the ashes of buried items forbidden? Tosfos explains that since the burned items have had the Mitzvah of burning done to them, their ashes are no longer forbidden. Items which are supposed to be buried remain forbidden forever.
The MINCHAS CHINUCH (145:5), YAD BINYAMIN, and others explain that Tosfos understands that the obligation to bury most items of Isurei Hana'ah was enacted by the Rabanan in order to prevent people from benefiting from the forbidden item (see Insights to Temurah 33:2 for two explanations of the words of Tosfos). The Avnei Milu'im explains that this applies to most items that are supposed to be buried. Chametz, however, is different. Since one who burns Chametz fulfills the Mitzvah to destroy Chametz before Pesach, when he burns the Chametz he has fulfilled the Mitzvah of "burying" Chametz. Just as Tosfos says that one may benefit from the remains of an item that has been burned because the Mitzvah has already been performed, one also may benefit from the ashes of Chametz once the Chametz has been burned, and, consequently, there is no reason to require their burial. (Y. MONTROSE)