Gemara

Once food was Mekabel Tum'ah, it does not cease to be Mekabel Tum'ah until it is improper for a dog;

Pesachim 5a (Beraisa) Suggestion: Perhaps "Ach ba'Yom ha'Rishon Tashbisu" (the Mitzvah to destroy Chametz) applies on Yom Tov itself!

Rejection (R. Akiva): Burning is an Av Melachah. (It is permitted on Yom Tov only for preparation of food).

5b (Rava): This shows that R. Akiva holds that Bi'ur Chametz must be through burning.

21a (Mishnah): As long as one may eat (Chametz on Erev Pesach), one may feed it to animals or sell it to a Nochri;

After the (permitted) time expires, it is Asur b'Hana'ah.

Objection: Obviously, one may sell it to a Nochri!

Bava Kama 41a - Question: What is the source that "its flesh will not be eaten" applies when a Shor ha'Niskal (an ox sentenced to be stoned) was slaughtered? Perhaps it forbids benefit when it was stoned, like R. Avahu taught (that "it will not be eaten" forbids even benefit).

Answer: If the verse came to forbid benefit, it should have said 'one will not benefit.'

Menachos 101a (R. Oshaya citing R. Shimon): If one was Mefagel a Minchah, it does not receive Tum'as Ochlim.

(Beraisa): The following Isurei Hana'ah receive Tum'as Ochlim - Orlah, Kilai ha'Kerem... and Basar v'Chalav.

R. Shimon says, they do not receive Tum'as Ochlim, except for Basar v'Chalav, for it had She'as ha'Kosher (it was once permitted).

(Rav Asi): He learns from "mi'Kol ha'Ochel Asher Ye'achel" that something is considered food (for Tum'as Ochlim) only if one may feed it to others.

Question (Beraisa - R. Shimon): A Pigul Minchah is Tamei.

Answer: The Beraisa discusses a Pigul Minchah that had She'as ha'Kosher. R. Oshaya discusses one that was never permitted.

(Beraisa - R. Shimon): One may benefit from a Treifah.

Rishonim

Rambam (Hilchos Ma'achalos Asuros 4:22): "V'Lo Yochal Es Besaro" teaches that a Shor ha'Niskal is Asur b'Hana'ah. One may not sell it or feed it to dogs or Nochrim.

Rashi (Pesachim 5b DH Shma): If R. Akiva held that one fulfills Bi'ur Chametz in any way, he would have no proof that the Mitzvah is not on Yom Tov, for on Yom Tov one could feed it to dogs or cast it to the sea.

Rashi (Menachos 101b DH Mah): A Terefah is Mutar b'Hana'ah, for the Torah said that one may give it to a dog. It is Hana'ah that one's dog eat, for he thinks (cares) about it.

Tashbatz (3:293): One may benefit from a Terefah. The Torah said that one may give it to a dog, and the Yerushalmi says that one may not give Isurei Hana'ah to a dog, even of Hefker.

Hagahos Maimoniyos (Hilchos Chametz 3:11): Ri ben Klonimus forbade to cast Chametz in a Hefker place. The Yerushalmi forbids feeding Chametz (after it is forbidden) even to dogs. If this refers to his own dog (this is obvious, for) Chametz is Asur b'Hana'ah! Rather, it forbids even other dogs. If one finds Chametz after it is forbidden, one may not leave it in the Chatzer for ravens to eat, for Bi'ur (eradication) is burning,

Or Some'ach (4:22): Ri ben Klonimus forbids feeding Chametz to Hefker dogs, and the Shulchan Aruch (448:6) rules like this. The Yerushalmi asked (like the Bavli) why the Mishnah needed to permit both feeding animals and selling to a Nochri. It answers that we infer that once it is Asur b'Hana'ah, both of these are forbidden. Had the Mishnah (explicitly) taught the Isur to feed animals after it is Asur, one might have thought this is only for his own animal. Why did we teach the Isur to give to a dog? It is forbidden even to a Nochri! Rather, it forbids even a Hefker dog. I.e. the Isur is due to Hana'ah, and it applies to all Isurei Hana'ah. However, the Tur's text of the Yerushalmi asks 'if we discuss one's own dog, he benefits from this.' This connotes that the Isur to feed a Hefker dog is not due to Hana'ah. Rather, we expound regarding Chametz "Lo Ya'achil (do not feed)." For other Isurei Hana'ah, it is permitted. This requires investigation.

Or Some'ach: The Tashbatz forbids feeding any Isurei Hana'ah even to a Hefker dog. I support this from Bava Kama 41a. If the verse came to forbid benefit, it should have said 'one will not benefit' and not "its flesh will not be eaten." If one may feed Isurei Hana'ah to a Hefker dog, we should say that "its flesh will not be eaten" forbids Hana'ah and feeding to dogs, for we expound Lo Ya'achil! However, Rashi says that if R. Akiva held that one fulfills Bi'ur Chametz in any way, one could feed it to dogs, even though R. Akiva forbids Hana'ah (32a). Rashi holds that the Yerushalmi forbids only regarding Chametz.

Note: Rashi in Menachos says that a Terefah is Mutar b'Hana'ah, for the Torah permits giving it to a dog. It is Hana'ah that one's dog eat, for he thinks (cares) about it.

Or Some'ach: In Menachos, the Gemara asked from a Beraisa in which R. Shimon says that a Pigul Minchah is Tamei. What was the question? If the questioner did not know that he admits when it once had Sha'as ha'Kosher, it should have asked about Basar v'Chalav! (Tosfos DH Pigel asked this.) Rather, the Gemara held that R. Shimon holds that if something is Asur to eat, it is as if physically it is not proper to eat, so it has no Tum'as Ochlim. He learns from an extra "Yochal". In Kerisus, we say that once something was Mekabel Tum'ah, it does not lose its Tum'ah until a dog cannot eat it. (This is like the Ra'avad and others; the Rambam holds that if it was Tamei, it is Tamei until a dog cannot eat it.) This is if it was once proper for people to eat, but not if from the beginning if was proper only for dogs. Orlah and Kil'ayim (and most Isurei Hana'ah) were never proper to feed to people, even to a Nochri he does not recognize. (Even giving for free is like a sale, for they will return the favor - PF.) Basar v'Chalav received Tum'ah (before they were mixed).

Maharil (Teshuvah 161, brought in Beis Yosef OC 448 DH v'Al): One may not feed another's animal, just like the Yerushalmi expounds "Lo Ya'achil" to forbid feeding a Hefker animal. The Agudah forbids feeding a Nochri's animal.

Chidushei Hagahos (4, on Tur, citing Maharalbach): This is because Chachamim forbade all benefit from Chametz b'Pesach. Even though he does not benefit, the animal benefits.

Poskim

Shulchan Aruch (OC 488:6): One may not feed Chametz b'Pesach even to another's animal or to a Hefker animal.

Gra (DH Asur): The Yerushalmi expounds "Lo Ya'achil" to forbid others' animals, i.e. of Nochrim. This implies that one may not feed a Hefker animal.

Bach (DH u'Mah): We cannot say that the Drashah forbids another person's animal, for he will benefit from the owner! (If Reuven fed Ploni's animal, Ploni will benefit Reuven.) Rather, it is Hefker. It is called 'of others' because others can acquire it.

Kaf ha'Chayim (104): The Levush says that Lo Ya'achil is an Asmachta (not a real Drashah mid'Oraisa).

Mishnah Berurah (26): We discuss one who found Chametz. Just like one may not feed his own animal, for this is Hana'ah, he may not feed a Hefker animal. Likewise, after Pesach one may not feed it Chametz that a Yisrael owned (and therefore became forbidden) during Pesach.

Mishnah Berurah (27): One gets Hana'ah through fulfilling his desire to satiate the animal. This is forbidden even if it is not his Chametz.

Shulchan Aruch (445:1): For Bi'ur Chametz, one may cast it to the sea. If it is hard Chametz and the sea will not break it up quickly, he crumbles it and then throws it.

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): 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.