Gemara

73a (Mishnah): The following apply to Ma'aser and Bikurim, but not to Terumah: one must bring them to Yerushalayim, one must recite Viduy over them, and they are forbidden to an Onen.

The Mishnah does not say that one may burn them (for benefit) if they are Tamei and that one is lashed for eating them if they are Tamei, which do not apply to Terumah! This shows that the Tana omitted cases.

73b - Question: What is the source that one may burn Tamei Terumah?

Answer #1 (R. Avahu): We learn from "I did not eradicate Mimenu (from it) in Tum'ah". You may not burn Tamei Ma'aser, but you may burn Tamei Terumah oil.

74a - Answer #2 (Rav Nachman): "V'Nosati Lecha Es Mishmeres Terumosai (plural)" alludes to two Terumos, Tahor and Tamei. It is "Lecha" (for your needs); you may cook with it while burning it.

Shabbos 25b - Answer #3 (Rav Nachman bar Yitzchak): "Titen Lo" (give Tahor Terumah which the Kohen himself can eat), do not give (Tamei Terumah, which will be fuel) for his fire;

This implies that the Kohen may use it for (fuel) for his fire.

25a (Rav): Just like it is a Mitzvah to burn Tamei Kodshim, it is a Mitzvah to burn Tamei Terumah. The Torah permitted to benefit at the time of burning (Terumah).

Pesachim 20b (Beraisa): If a barrel of Terumah oil broke and it is about to fall into Tamei Chulin below, R. Eliezer and R. Yehoshua agree that if one can save a Revi'is in Taharah, he must do so. If he cannot, he must allow the Terumah to fall. He may not actively Metamei it. They argue about wine.

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

This is an important usage. Shmuel said that it is appropriate to spend one Sela for a Log of wine for drinking, and two Sela'im for a Log for Ziluf!

Answer: R. Yehoshua permits to Metamei new wine, for (this would be a big loss. It has no scent, so) it is not fitting for Ziluf.

One can put oil in a disgusting Keli, so we are not concerned for Takalah (lest someone come to consume it). If one put wine in such a Keli, it would not be fit for Ziluf. One may not let wine age in a clean Keli, due to Takalah!

23a: A Zar may benefit from Terumah:

(Mishnah): One may make an Eruv on behalf of a Nazir with wine, and on behalf of a Yisrael with Terumah.

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

Temurah 33b (Mishnah): The following must be burned: Chametz during Pesach, Tamei Terumah, Orlah, and Kil'ei ha'Kerem;

What burns is burned. What is buried (because it does not burn) is buried.

We burn bread and oil of (Tamei) Terumah (and benefit from the fire).

Rishonim

Rif (Shabbos 12a): "...Mishmeres Terumosai" discusses two Terumos, Tahor and Tamei. Just like you may not benefit from Tahor Terumah before separating it, you may not benefit from Tamei Terumah before separating it.

Rambam (Hilchos Terumos 2:14): A Kohen may benefit from Tamei Terumah while burning it. He may light (Tamei Terumah oil) or use grain to ignite his oven.

Mishneh l'Melech: Even though Terumah is not Asur b'Hana'ah to Zarim, they may not benefit from it in a way that consumes it. This is even if the Zar owns it, e.g. he inherited it from his mother's father (a Kohen).

Rambam (12:12): If Terumah bread became Tamei, one casts it among wood until he burns it. If Terumah oil became Tamei, he puts it in a disgusting Keli so no one will eat it. If liquids that do not burn, such as wine, became Tamei, they are buried.

Ri Korkus: This is like the Mishnah in Temurah. Even though Beis Hillel permit Ziluf of wine in a Beraisa, this is because they are not concerned for Takalah. In many places we are concerned for Takalah, so the Rambam followed the Mishnah. The Rambam did not accept the resolution of Tosfos (Pesachim 20b DH Ra'uy, that 'we bury what is buried' does not refer to wine), for if so it should have said 'what is used for Ziluf we use for Ziluf.'

Rashi (Pesachim 34a DH v'Hayu): They used to buy cheaply Tamei Terumah wheat for Aba Sha'ul from Kohanim.

Tosfos (DH Mechamin): This was for Kohanim of Rebbi's house. Hana'as Kiluy (benefit from Terumah that consumes it) is forbidden to Yisraelim.

Pnei Yehoshua: Rashi connotes that it was for Rebbi himself. The Terumah was from Chutz la'Aretz; Chachamim were not stringent about Hana'as Kiluy of Terumas Chutz la'Aretz. Alternatively, they were lenient about Terumah of Eretz Yisrael that became Tamei mid'Rabanan.

Question (Hagahos Baruch Ta'am): If so, the Terumah is Tahor mid'Oraisa, so Hana'as Kiluy is forbidden!

Mabit (2:198): Rashi permits even a Yisrael to use Tamei Terumah wine for Ziluf.

Rebuttal (Ma'adanei Aretz 2:14:4 DH Gam): It is difficult to say that Rashi permits Hana'as Kiluy to a Yisrael. Perhaps this is only for Terumah nowadays, which is mid'Rabanan.

Poskim

Shulchan Aruch (YD 331:19): Nowadays the Terumah we separate must be burned.

Rema: A Kohen may leave around Tamei Terumah until he will benefit from burning it. A Yisrael may not benefit from burning it unless a Kohen benefits at the same time. A Yisrael may benefit from it himself in a way that does not consume it.

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.

Rema: A Kohen may benefit from Tamei Terumah only if received Hechsher (was wet intentionally), for then it can become Tamei and may be burned. It is good to be Machshir produce before Miru'ach (final processing), so he can Metamei it and benefit from the Terumah.

Gra (50): The Yisrael may be Metamei it because this does not cause a loss to the Kohen.

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 until he will burn it later. Why is he concerned for transgression when it was not Huchshar before Miru'ach?

Answer #1 (Bach DH u'Mah she'Chosav v'Tov): If it was Huchshar before Miru'ach, the owner himself will burn it soon. (He will not forget that it is Terumah.) If it was not Huchshar beforehand, he must bury it, and perhaps someone else will find it and use it.

Rebuttal (Taz 9): One may keep Tamei Terumah as long as he wants!

Answer #2 (Taz): The text of the Tur should not say 'immediately'. Tamei Terumah is put in a special Keli (to burn it later), so we are not concerned for mistakes. One does not leave Tahor Terumah in a special Keli, so we are concerned for Takalah.

Answer #3 (Prishah 18): People know that nowadays Terumah (is normally Teme'ah and) is burned. They will not eat it. When they see Tahor Terumah that may not be burned, they realize that it is unlike other Terumah, and might eat it.

Answer #4 (Drishah 5): We are more stringent about Tahor Terumah because one may not benefit from it in any case.

Shach (35): If Miru'ach was before Hechsher, Terumah must be separated. Therefore he may not be Machshir it afterwards, for this causes Tum'ah to Terumah.

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.