Gemara

15a (Mishnah - R. Eliezer): If a barrel of Terumah wine broke, and it is about to fall into Tamei Chulin below, if one cannot save a Revi'is in Taharah, he must allow the Terumah to fall (even though it will mix with the Chulin, become Tamei and forbid it all even to a Kohen). He may not actively be Metamei it (e.g. catch it in a Tamei vessel);

R. Yehoshua says, (in order to save the Chulin) he may actively be Metamei the Terumah.

20b (Beraisa): They argue about wine. Regarding oil, all forbid to actively be Metamei the Terumah in order to save the Chulin.

Question: Presumably, R. Yehoshua forbids regarding oil, for even Tamei Terumah oil may be burned. (It is not a total loss.) Likewise, one may use Tamei Terumah wine for Ziluf (sprinkle it to scent the house)!

Answer: One can put oil in a disgusting Keli, so we are not concerned for Takalah (lest someone come to consume it). New wine has no scent, so it is not good for Ziluf. If one puts it in a filthy Keli, it would not be proper for Ziluf. One may not let it age in a clean Keli, due to Takalah!

34a (R. Avin bar Rav Acha): Aba Sha'ul was the kneader for Rebbi's house. They would use wheat of Tamei Terumah to heat water for him for kneading.

Question: We should be concerned for Takalah!

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

Temurah 33b (Mishnah): The following are Nisrafim (must be burned):

Chametz during Pesach, Tamei Terumah...

What it is normal to burn is burned. What it is normal to bury (but cannot be burned) is buried.

Kerisus 9a (Beraisa): Nowadays, a convert must set aside a quarter-Dinar to buy birds [to offer immediately when the Mikdash will be rebuilt].

R. Shimon says, R. Yochanan ben Zakai repealed this, lest one benefit from the money [and transgress Me'ilah].

Sukah 35b: One may not take an Esrog of Tahor Terumah for the Mitzvah. R. Ami and Rav Asi argue about whether this is because he is Machshir it, or because he causes a loss. (Handling the Esrog harms the peel.)

Rishonim

Rambam (Hilchos Terumos 2:14): A Kohen may benefit from Tamei Terumah while burning it.

Ri Korkus: One may give Shemen Sereifah (Terumah oil that must be burned) to any Kohen, because it has no Kedushah. We are not concerned if he is Metamei it. There is no concern for Tamei Terumah.

Rashi (33b DH Es): Regarding Orlah and Kil'ai ha'Kerem, we burn solids and bury liquids.

Rashi (25a DH Mitzvah): It is a Mitzvah to burn Tamei Terumah, similar to Kodesh. Also, it is to avoid Takalah.

Tosfos (DH Ken): According to Rashi's second reason, Bi'ur (eradication) of Tamei Terumah is not only through burning. Rashi explicitly said so in Eruvin. I disagree. In Temurah, Tamei Terumah is listed among Nisrafim (Isurim that must be burned). We learned that Nisrafim may not be buried! Rather, mid'Rabanan it must be burned, like Tamei Kodesh. Alternatively, this is mid'Oraisa, since it is called Kodesh.

Question: Why do we permit (Pesachim 20b) using Tamei Terumah for Ziluf? The Mishnah in Temurah teaches that what is [normally] buried must be buried!

Answer (Tosfos): The Mishnah teaches that liquids of Orlah and Kil'ai ha'Kerem must be burned, but not Terumah, for it is Mutar b'Hana'ah [to Kohanim].

Tur (YD 331): Even though in Pesachim we say that we are concerned for Takalah unless he puts it in a repulsive Kli, nowadays that no Terumah is eaten, we are not concerned.

Bach (8): This is from Sefer ha'Terumah.

Gra (42): This is difficult. Why is this different than Hekdesh? In Yoma (66a), we say that nowadays, we are Oker Hekdesh (lock it up to die). A convert does not separate money for the birds he must bring when the Mikdash will be rebuilt. In Pesachim (20b and 34a), the only Heter found to benefit from Tamei Terumah is if it was made repulsive.

Note: Perhaps Hekdesh is different, because in any case we cannot benefit from it. What is the question from Pesachim? In those days, Tahor Terumah was eaten!

Poskim

Shulchan Aruch (YD 331:19): Nowadays we burn Terumah Gedolah, due to Tum'ah.

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. The Kohen burns it.

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).

Rema: 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.

Tur: One may not be Metamei Tahor Terumah. Rather, he buries it. It is better to be Machshir the produce before Miru'ach, so he can be Metamei it and burn it immediately, to avoid possible transgression.

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 Takalah, 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 all know 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 (Bach DH u'Mah she'Chosav v'Tov): Above, he may burn it immediately, so we are not concerned for Takalah. Here, if he would not be Machshir it, it would be a long time until he buries it. 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 #3 - 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.'

Taz (8): He can save it to burn it later, even in a clean Keli. Nowadays no Terumah is eaten, so Kohanim designate a special place for Terumah, so there is no concern lest it be eaten.

Gra (50) and Shach (35): One may not be Machshir it after Miru'ach, for then one may not cause Tum'ah to it, for one must separate Terumah. We learn from Sukah 35b. The Yisrael may be Metamei it before Miru'ach, for this does not cause a loss to the Kohen.

Gra (50): Sefer ha'Terumah was unsure whether nowadays [that all Terumah is mid'Rabanan] one may be Machshir even after Miru'ach, to prevent Takalah.

Minchas Yitzchak (4:13): Perhaps the law did not change regarding Chalah of Chutz la'Aretz, which is more lenient. (Since the Takalah is smaller, perhaps one may not be Machshir it - PF.)

Gra (48): Even if a Safek Tum'ah occurred, one may not burn it. There are countless sources for this. This is unlike Chametz Terumah, which one burns on Erev Pesach. Since it must be burned, it is like Tamei Terumah.

Kapos Temarim (Sukah 35b DH Chad): Rashi explains that one opinion forbids using an Esrog of Terumah because he is Machshir it. Why does the other opinion disagree? All forbid to be Metamei Tahor Terumah! Perhaps the other opinion holds that the Isur is to be Metamei, but one may be Machshir, for this merely causes Tum'ah to Terumah. Alternatively, also he forbids to be Machshir. He holds that one can be careful not to be Machshir, e.g. not to take the Esrog together with a wet Lulav. The Rambam (Hilchos Lulav 8:2) rules like the opinion that forbids because he is Machshir. In Hilchos Terumos (12:1) he forbids to be Metamei Terumah. Why didn't he teach that one may not even be Machshir, for this causes Tum'ah to Terumah? This requires investigation. The Bartenura (Parah 11:7) says that one may not sprinkle Mei Chatas with Ezov of Tahor Terumah, for this ruins it. He is not concerned for Hechsher. The Rambam did not explain the reason for the Mishnah. It makes a difference. If it is due to Hechsher, one may use Terumah that was already Huchshar, like Tosfos says! This requires investigation.

Minchas Yitzchak (4:13): Beis Yitzchak (OC 105:5) says that this depends on whether Hechsher is the beginning of Tum'ah. Rav and R. Yochanan argue about this (Chulin 118b). The Rambam rules like R. Yochanan, that it is the beginning of Tum'ah. (Therefore one may not be Machshir Terumah.) I say that nowadays that we are all Tamei, this is not relevant. Hechsher Vadai (immediately) makes it Tamei, for one must touch it.

Seder Hafrashos Terumos u'Ma'aseros (based on Chazon Ish): Nowadays one may not eat Terumah Gedolah or Terumas Ma'aser. Therefore, if they were Tamei, i.e. one of the seven liquids came on them after they were detached and one touched them, he must burn them. He may not benefit from them at the time of burning, unless a Kohen benefits with him. If Terumah did not become Tamei, he buries it or wraps it in paper and then throws it in the trash. One may not act disgracefully with Terumah after it was separated, or be Metamei it.

Minchas Yitzchak (4:13): The Chazon Ish permitted to wrap and throw out Tahor Terumah, even though it should be buried. The custom is to do so even for Tamei Terumah. Perhaps Chachamim authorized this, for it is relevant to many people in Eretz Yisrael. Mostly women deal with this, and they are not experts to distinguish between whether or not it was Huchshar. The Isur to burn Tahor Terumah is more stringent than [the Mitzvah of] burning Tamei Terumah (i.e. if people think that Terumah is Tamei, but perhaps they err, it is better that they not burn it - PF), for Rashi holds that the Mitzvah of Bi'ur Tamei Terumah is not limited to burning. Tosfos holds that it must be burned, but perhaps this is only mid'Rabanan, even for Terumah mid'Oraisa.

Minchas Yitzchak (4:13:10): I do not understand why nowadays even Yir'ei Shomayim in Eretz Yisrael do not burn Terumah. They merely wrap it up (to avoid disgrace) and throw it in garbage bins. Most Poskim require burning it! Perhaps there is more room to be lenient about Chalah of Chutz la'Aretz, which has no source mid'Oraisa.