Gemara

(Beraisa - R. Yehudah): A Kal va'Chomer obligates burning Chametz;

Bal Yera'eh and Bal Yimatzei do not apply to Nosar, yet Nosar must be burned. They apply to Chametz, so all the more so one must burn it!

Objection (Chachamim): Your stringency leads to a leniency! If one has no wood, you exempt him from Bi'ur. "Tashbisu Se'or mi'Bateichem" obligates Bi'ur in any way you can!

R. Yehudah: Rather, "Lo Sosiru" applies to [leaving over] Chametz, and to Nosar. Just like Nosar must be burned, also Chametz!

Chachamim: According to you, Asham Taluy disproves this. Bal Sosiru applies to it. We say that it is burned, but you say that it is Nikbar (buried)!

5a (Beraisa - R. Akiva): "Ach ba'Yom ha'Rishon Tashbisu" cannot apply on Yom Tov itself, for burning is an Av Melachah.

(Rava): We learn from R. Akiva that Bi'ur Chametz must be through burning.

21a (Mishnah - R. Yehudah): Bi'ur is only through burning;

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

28a - Question: Does "crumble" apply only to throwing into the wind, or even to casting into the sea?

Answer #1 (Rabah): One may cast Chametz to any river (not only the Dead Sea), so it must be crumbled [lest someone in a boat find it].

Answer #2 (Rav Yosef): Chametz dissolves in water, so it need not be crumbled!

Question (Beraisa #1): If one was in a boat [at the time of Bi'ur], he crumbles it and casts it into the sea.

Answer: Wheat is unlike bread.

Temurah 33b (Mishnah): 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, for ashes of Nisrafim are permitted, but ashes of Nikbarim are forbidden.

Rishomin

Rambam (Hilchos Chametz 3:11): Bi'ur Chametz is through burning, or crumbling and throwing to the wind, or casting to the sea. If it was hard Chametz, and the sea will not break it up quickly, he crumbs it and then throws it to the sea.

Rosh (2:3): The Halachah follows Rabah. Rashi rules like R. Yehudah, for the Stam Mishnah at the end of Temurah is like him. R. Yehudah was silent (did not answer Chachamim's question), but we are not silent. Also Rav Bartzeloni rules like him. Rava derived that Chametz must be burned. The Ge'onim rule like Chachamim. Ba'al ha'Itur and R. Yonah say so. Even though there is a Stam Mishnah in Temurah, Ein Seder l'Mishnah. (Perhaps Maseches Pesachim, in which Tana'im argue about this, was taught after Temurah. If so, the Halachah need not follow the Stam Mishnah.) Therefore, the Halachah follows the majority. Why don't Rabanan learn from Nosar? According to them, there is no question! I answer that they can say that Nosar is Kodesh, therefore it is proper to be stringent to burn it. Burning is a stringency (Temurah 34a). [R. Yehudah holds that] one may be stringent to burn Nikbarim. R. Tam (12b DH Eimasai) explains that at the time of Bi'ur, one may do Bi'ur in any way, i.e. in the sixth hour, when people normally do Bi'ur. According to him, it does not matter so much whom the Halachah follows, for one does not [normally] delay Chametz past six hours. However, Rashi holds that R. Yehudah agrees that in the fifth hour one may do Bi'ur in any way.

Poskim

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

Kaf ha'Chayim (1): These correspond to three ways the Mitzriyim killed Yisre'elim. They burned Bechoros, choked (this is like crumbing) newborn babies, and cast them to the river. This helps us remember the slavery, and thank Hash-m for freeing us.

Beis Yosef (DH uvi'Gemara): The Rosh rules like Rabah, but the Rambam rules like Rav Yosef, who requires crumbling hard things. He explains "wheat is unlike bread", i.e. one need not crumb fermented wheat. Even if it is gathered, the water will scatter it. One must crumb bread, for the water will not cut it. This is unlike Rashi. The Tur wrote Stam like the Rosh. He did not distinguish the Dead Sea from other seas, for normally people do Bi'ur where others are found, therefore even the Dead Sea is like other rivers.

Magen Avraham (1): Our texts of the Gemara say that Chachamim permit even to crumble it. The texts of the Rif and Rosh connote that 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): The Rambam forbids burning only the Nikbarim listed in the Mishnah. This excludes Chametz, for Chachamim explicitly taught that it may be burned or buried. Therefore, the Rambam did not list it among Nisrafim and not among Nikbarim. It is better to be stringent to burn it, lest people transgress Bal Yera'eh and Bal Yimatzei, even though this is a leniency for the ashes. Also, the ashes are forbidden only if one did not do Bi'ur in the proper time. This is rare, so Chachamim did not decree about it. The Magen Avraham is difficult.

Gra (DH Sorfo): Even according to the texts that do not say 'even', Chachamim permit burning, for R. Yehudah says 'only burning.' Also, we cannot say that all the Mishnayos and Beraisos that mention burning Chametz are [attributed to] R. Yehudah, and R. Meir, R. Yosi, R. Shimon, R. Gamliel, R. Eliezer, R. Yehoshua and R. Elazar Ish Bartusa all agree. (We would not call this [only] R. Yehudah's opinion!) Rather, Chachamim agree that the primary Mitzvah is burning.

Taz (1): Why does the Shulchan Aruch rule like the Rambam, who rules like Rav Yosef? The Halachah follows him against Rabah only in three places! (This is not one of them.) The Tur did not distinguish between hard and soft Chametz, i.e. like Rabah, and wrote "and so wrote the Rambam." I say that the Rambam rules like Rabah. The Dead Sea breaks up Chametz quickly. Rav Yosef holds that wet bread is repulsive, but wet wheat is not. The Rambam said that it depends on whether the sea will break up the Chametz quickly, for it depends on the sea and the Chametz. The Tur did not distinguish different seas, for their natures are not uniform. He is stringent to require crumbing in every case.

Gra (DH Oh): He rules like Rav Yosef, because it says "Amar Lei Rav Yosef", and not "Rav Yosef Amar." The Gemara says so in many places [to show that the Halachah follows the latter opinion].

Damesek Eliezer: "Amar Lei..." connotes that this is a clear reasoning.

Mishnah Berurah (1): One burns it until it becomes coals.

Mishnah Berurah (2): He crumbs it so finely that no finder will benefit from it.

Mishnah Berurah (3): It suffices to throw it to a river.

Shulchan Aruch (ibid.): If it is hard Chametz and the sea will not break it up quickly, he crumbles it and then throws it.

Gra (DH ha'Chametz): Wheat is hard Chametz, unlike Rashi says.

Mishnah Berurah (4): Hard Chametz is hard dry bread or dry fermented wheat.

Magen Avraham (2): If one casts it into a latrine, he need not crumble it. We need not be concerned lest a dog or pig eat it, for if so we should be concerned also in the sea, lest fish eat it! Rather, we are not concerned because he does not overtly feed them. If one has gathered water with fish inside, he may not throw the Chametz there, for he benefits from it.

Mishnah Berurah (5): Once it was cast into a latrine, no person can benefit from it. If a pig is there at the time, he may not cast the Chametz to the pig. If the pig will come later, we are not concerned.

Kaf ha'Chayim (10): People who leave Chametz at the side of the road in the sixth hour do improperly. One must exterminate it from the world! However, we are among Nochrim who mock us for casting Chametz on the road, so one should not do so even before the sixth hour, due to Chilul Hash-m. Rather, one gives a gift to a Nochri, covertly casts it to a river, or burns it in his house.

Rema: The custom is to burn it.

Taz (2): The custom to burn it is due to different opinions about whom the Halachah follows. Therefore, we are stringent. If one fulfills Bi'ur Chametz through burning after six hours, and afterwards found Chametz, he need not burn it. It suffices to destroy it, for after six hours [on Erev Pesach] we are lenient about Chametz like for other Isurim, e.g. [we do not forbid a mixture due to] any amount (of Chametz, unless it gives taste), or if it harms the taste. Even during Pesach we hold like Rabanan, just it is a custom to burn like R. Yehudah.

Gra (DH veha'Minhag): I.e. even R. Yehudah requires burning only after six hours, like R. Tam, for he learns from Nosar. The Yerushalmi says so. Since we burn it before the Torah forbids it, it is a mere custom to burn it. The custom is like Rashi, who says that R. Yehudah requires burning in the sixth hour, and permits in any way before this. All the Poskim hold like R. Tam.

Gra (DH v'Tov): It is a stringency to burn it. R. Yehudah learns from Nosar, which is burned only during the day, and not on Yom Tov. He holds that also Chametz is burned only during the day.

Mishnah Berurah (6): The custom is to burn even during the time when it is proper (permitted to benefit from it), i.e. the fifth hour on Erev Pesach, and all the more so after six hours or during Pesach. Then surely one should be stringent to burn it.

Kaf ha'Chayim (11): Also, Chametz alludes to Klipos (shells of Tum'ah). Burning alludes to removing Tum'ah and approaching the time of redemption.

Kaf ha'Chayim (12): The Rema and Maharshal require burning even before the sixth hour.

Kaf ha'Chayim (14): It is good to burn with Aravos [that were used for the Mitzvah].