Gemara

(Beraisa - R. Yosi): We learn the power of Chazakah from "Bnei ha'Kohanim... they could not find their lineage documents, and they were banished from Kehunah. Ha'Tarshasta (Nechemyah) told them, they will not eat Kodshei Kodoshim until the Urim v'Turim are restored (when Moshi'ach comes)."

He told them that since they had a Chazakah (in Bavel) to eat Terumah, they may continue to eat Terumah in Eretz Yisrael.

Question: How does this show the power of Chazakah?

Answer #1: In Bavel, they only ate Terumah mid'Rabanan. In Eretz Yisrael, they may eat Terumah mid'Oraisa.

Answer #2: Also in Eretz Yisrael, they are only permitted Terumah mid'Rabanan. Even though in Eretz Yisrael, there was concern lest they come to eat Terumah mid'Oraisa, due to the power of Chazakah, no decree was made.

Chulin 130b (Beraisa): "V'Zeh Yiheyeh Mishpat ha'Kohanim" - teaches that Beis Din enforces the Matanos.

This means that Beis Din forces a Yisrael to give Matanos to a learned Kohen.

(Rav Shmuel bar Nachmani): We do not give Matanos to an ignoramus Kohen - "Lema'an Yechezku b'Soras Hash-m".

Pesachim 34a (R. Avin bar Rav Acha): In Rebbi's house, they used wheat of Tamei Terumah to heat water for Aba Sha'ul, who kneaded in Taharah.

Question: We should be concerned for Takalah (lest one eat Tamei Terumah)!

Answer (Rav Ashi): He cooked it and made it repulsive.

Rishonim

Rambam (Hilchos Terumos 6:2): Only a Kohen Meyuchas may eat mid'Oraisa Terumah. Kohanim of Chazakah may eat only Terumah mid'Rabanan. Tahor Terumah is given only to a Kohen Chacham, for one may not eat Tamei Terumah, and Amei ha'Aretz are Muchzakim to be Temei'im. Therefore, one may give Tamei Terumah to any Kohen.

Rebuttal (Ra'avad): This is not clear.

Kesef Mishneh: The Mishnah in Chalah (4:9) lists Shemen Sereifah (Tamei Terumah oil) among gifts that are given to any Kohen. The Rambam explains that this means to a Chaver or an Am ha'Aretz. The Ra'avad explains like R. Shimshon, that it may be given to any Chaver, even if he does not eat Chulin in Taharah. The Yerushalmi is like the Rambam. However, why aren't we concerned lest a Kohen Am ha'Aretz eat it b'Tum'ah?

Ri Korkus: One may give Shemen Sereifah to any Kohen because it has no Kedushah. We are not concerned if he is Metamei it. Tosfos (Chulin 130b DH Minayin) says that one may give it even to an Am ha'Aretz, but only when no Chaver is around. Other gifts may not be given to an Am ha'Aretz even if no Kohen is around. This is difficult. Some say that there is no Isur to give to an Am ha'Aretz, just it is improper. There is no concern for Tamei Terumah.

Poskim

Shulchan Aruch (YD 331:19): Nowadays we burn Terumah Gedolah, due to Tum'ah. Therefore, the Shi'ur is any amount.

Rema: Nowadays one may give it to any Kohen, a Chaver or Am ha'Aretz, even if he is not Meyuchas, and he is just Muchzak to be a Kohen.

Gra (38): Even though it is a small amount and it is Tamei, one must give it to a Kohen (and not burn it oneself).

Gra (41): The Tur holds that the Isur (to Zarim) of Hana'ah that consumes the Terumah is mid'Rabanan, therefore the Kohen need not be Meyuchas. Tosfos holds that the Torah forbids such Hana'ah.

Rema: He burns it. He may leave it and burn it (later), for he may benefit from it at the time it burns. A Zar may benefit from it at the time it burns only if a Kohen benefits with him. Other Hana'ah that does not consume it is permitted even to Zarim. This refers to Terumah that was Huchshar to become Tamei, for then it is Tamei and one may burn it. If it was not Huchshar, it is Tahor, and one may not burn it. It is good to be Machshir the produce to receive Tum'ah before Miru'ach (final processing), so the Terumah will be Tamei and one may burn it.

Question (Beis Yosef DH u'Mah she'Chosav v'Yisrefenah): Above, the Tur permits to leave Tamei Terumah and burn it. We are not concerned, for nowadays no Terumah is eaten. Here, he says that we bury Tahor Terumah, but it is better to be Machshir the produce before Miru'ach, and burn the Terumah Immediately! It is unreasonable to say that here is different, for it was initially Tahor and he was overtly Machshir it to receive Tum'ah.

Answer #1 (Prishah 18): When it was initially Tamei and destined to be burned, we are not concerned lest one benefit from it, since all know that nowadays Terumah is burned. Here, it was Tahor and forbidden to burn it. Not everyone knows that it became Huchshar. Perhaps people will err to think that it is unlike regular Terumah, and they will come to eat it. Therefore, the Tur suggests to be Machshir it and burn it immediately.

Answer #2 (Drishah 5): Above, one may benefit from burning it, therefore Chachamim were not stringent to obligate burning it immediately. Here, one may not benefit from it, so they are stringent.

Answer #3 (Bach DH u'Mah she'Chosav v'Tov): Above, he may burn it immediately, so we are not concerned lest one come to eat it. Here, if he would not be Machshir it, it would be a long time until he buries it, and perhaps someone else who does not know that it is Terumah will eat it. It does not help to put grain in a repulsive Kli, for it will not become repulsive.

Rebuttal (and Answer #4 - Taz 9): One may keep Tamei Terumah as long as he wants! Rather, 'immediately' is a printing mistake in the Tur. One normally puts Tamei Terumah in a Tamei Kli special for it, so we are not concerned lest one eat it. He may not put Tahor Terumah in that Kli, so we are concerned. Therefore, it is better that the Terumah be Tamei from the start. Also the Rema does not mention 'immediately.'

Question (Gra 42): Why is Tamei Terumah different than Hekdesh nowadays, about which we are concerned for Takalah?! In Pesachim, we say that they cooked it make it repulsive!

Gra (50) and Shach (35): One may not be Machshir it after Miru'ach, for then one is obligated to separate Terumah. Sefer ha'Terumos was unsure whether one may do nowadays, to prevent Takalah.