Gemara

23b (Mishnah): One may not use Shemen Sereifah (Tamei Terumah oil).

Question: What is the reason?

Answer #1 (Rabah): Because it is a Mitzvah to burn it, we decree lest he tilt the Ner.

Answer #2 (Rav Chisda): We forbid only when Erev Shabbos is Yom Tov, for we may not burn Kodshim on Yom Tov.

24b (Mishnah): One may not burn Shemen Sereifah (Tamei Terumah) on Yom Tov.

This is because we do not burn [Pasul] Kodshim on Yom Tov.

Question: What is the source of this?

Answer #1 (Chizkiyah): The repetition "Lo Sosiru Mimenu (do not leave over meat of the Pesach) Ad Boker veha'Nosar Mimenu Ad Boker ba'Esh Tisrofu" teaches that Nosar is burned on another morning (after Yom Tov, i.e. Nisan 16).

Answer #2 (Abaye): One may burn on the Mizbe'ach "Olas Shabas b'Shabato", but not (limbs of) an Olah of Chol (Erev Shabbos or Erev Yom Tov) on Shabbos or Yom Tov. (All the more so, one may not burn Pasul Kodshim on Yom Tov.)

Answer #3 (Rava): "Hu Levado Ye'aseh Lachem" (Melachah of food preparation is permitted on Yom Tov), but not circumcision after day eight. A Kal va'Chomer would have permitted this. (There is no fixed time to burn Kodshim. All the more so it does not override Yom Tov.)

Answer #4 (Rav Ashi): "Shabaton" is a Mitzvas Aseh to rest on Yom Tov (in addition to the Lav of Melachah). The Aseh to burn Kodshim does not override a Lav and an Aseh.

Inference: On a Yom Chol one may burn [and benefit from] Shemen Sereifah (and benefit from this)!

(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).

Question: Where does the Torah permit this?

Answer #1 (Rav Nachman): "...Mishmeres Terumosai" discusses two Terumos, Tahor and Tamei. "Lecha" teaches that they are yours (the Kohen's. You may not eat Tamei Terumah, but) you may burn it and cook over the fire.

25b - Answer #2 (Rav Nachman bar Yitzchak): "Titen Lo" (give (Tahor) Terumah, which the Kohen can eat), but not (Tamei Terumah, which will be fuel) for his fire;

Inference: The Kohen may use it for (fuel) for his fire.

26a (Beraisa): One may not light with Tamei Tevel (untithed produce) on Chol, and all the more so not on Shabbos.

"...Mishmeres Terumosai" discusses two Terumos, Tahor and Tamei. Just like you may not benefit (eat) from Tahor Terumah before separating it (it is Tevel), you may not benefit (e.g. burn) from Tamei Terumah before separating it.

Pesachim 32a (Beraisa #1 - R. Akiva): If a Zar ate Chametz of Terumah on Pesach he is totally exempt [because one pays according to the value];

R. Yochanan ben Nuri obligates (because one pays according to the volume).

R. Yochanan ben Nuri: This is like Tamei Terumah. One may never benefit from it, yet one pays for eating it!

R. Akiva: One may benefit from Tamei Terumah while burning it. Chametz b'Pesach is Asur b'Hana'ah. It is like Tamei Terumah of grapes. One may not eat them, and they do not burn!

Rishonim

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

Tosfos (24b DH Lefi): How can we learn Terumah from Kodshim? One may not benefit from Kodshim, so he burns not for the need of Yom Tov. One may benefit from burning Tamei Terumah, like from Chulin. One can't one burn it on Yom Tov for benefit [e.g. cooking]? Why is it worse if he does another Mitzvah at the time (burning Tamei Terumah)? The Riva says that it is worse. The Torah permitted only Hana'ah of burning it. He burns it for the Mitzvah of burning it, and not for the Hana'ah. This is like the opinion that forbids voluntary Korbanos on Yom Tov. He holds that the Torah permits eating meat of Kodshim, like guests at Hash-m's table. They are slaughtered primarily for Hash-m. The Ri disagrees. There, we do Avodah with the Korban, unlike Tamei Terumah that we burn. Rather, Chachamim decreed not to burn Tamei Terumah on Yom Tov, lest one burn Kodshim on Yom Tov.

Maadanei Aretz (4, DH mi'Pashtos): The Rambam connotes that it is a Mitzvah to benefit from Tamei Terumah while burning it. In Perek 12 (below) he connotes that one may not ruin Tamei Terumah if one could benefit from it. Amtachas Binyamin (Terumos 16) learns from here that the Rambam holds that mid'Oraisa, one must burn Tamei Terumah. If so, it is like Chulin. Why can't one burn it on Yom Tov for cooking? Tosfos holds that one may benefit from burning Tamei Terumah, but this is not a Mitzvah. The Rambam holds that benefit is a Mitzvah, so it is hard to give Tosfos' answer, that the Mitzvah of burning is primary. I answer that the Rambam holds that also [benefit from] Tamei Terumah is permitted from Hash-m's table, so it is burned primarily for Hash-m.

Rambam (12:1): One may not be Metamei Terumah of Eretz Yisrael, just like Kodshim, not bring it to be Tamei. One may not ruin it. Rather, one etas Tahor Terumah, and burns Tamei Terumah.

Rambam (Hilchos Yom Tov 3:8): One may not burn Tamei Chalah on Yom Tov, for on Yom Tov we do not burn Kodshim that became Tamei. Burning Kodshim is an Aseh - "ba'Esh Tisaref." A Lav and an Aseh forbid Melachah on Yom Tov not for the sake of food. An Aseh is not Docheh a Lav and an Aseh.

Tosfos (24b DH v'Lo): The Ri asked about why Rava didn't answer like Rav Ashi, that an Aseh is not Docheh a Lav and an Aseh. In Beitzah (8b), Rava agrees that Yom Tov is also an Aseh, so he needed to give another answer there.

Or Some'ach: Bris Milah overrides the Lav and Aseh of cutting off Tzara'as, and the Aseh for a Kohen Metzora and a Nazir to shave overrides the Lav and Aseh of shaving the beard and cutting the Pe'os. Tosfos (Yevamos 5a DH she'Chen) says that we do not learn from here, for it is like Shenei Kesuvim (two verses, one of which teaches something which could have been learned from the other). If so, according to R. Yehudah, who learns from Shenei Kesuvim, an Aseh is Docheh a Lav and an Aseh! This answers the question of Tosfos (24b). Rava agrees that Yom Tov is also an Aseh, but he wanted to answer also for R. Yehudah (why one may not burn Tamei Terumah on Yom Tov). For R. Yehudah, the Aseh is Docheh the Lav and Aseh of Yom Tov.

Rambam (Hilchos Pesulei ha'Mukdashim 19:5): One may not burn Pigul or Nosar at night. One may not burn Pigul, Nosar or Tamei on Shabbos or Yom Tov.

Question (Rashba 26a DH u'Tmiha): Why does the Beraisa say 'and one need not say so about Shabbos'? What addition Isur is there on Shabbos? We do not burn Kodshim on Shabbos, but this answers only for the opinion that Tevel (Peros from which Terumah was not yet separated) is like Terumah. One opinion holds that it is unlike Terumah. How can he answer? This requires investigation.

Answer #1 (Oneg Yom Tov 50 DH uv'Zah): On Shabbos there is Mitzvah ha'Ba b'Aveirah. He fulfills the Mitzvah of Ner Shabbos through an Isur Hana'ah of consuming Tevel. Even on Yom Tov Sheni, which is mid'Rabanan, Mitzvah ha'Ba b'Aveirah disqualifies (Sukah 30a). Mitzvos Lav Leihanos Nitnu does not apply, for the Mitzvah of Ner Shabbos is for the sake of Hana'ah and Oneg.

Answer #2 (Sefas Emes 26a DH Ein): One must light Ner Shabbos. This is absolute Hana'ah. If not for the Tamei Tevel, he would need to buy other oil. There is less Hana'ah in a weekday, for then he could sit in the dark. His benefit is only that he can see. Me'ilah does not apply to sights. Why did the Rashba suggest that Tevel is like Terumah, and it is due to the Isur to burn Kodshim on Shabbos? We concluded (23b) that we are not concerned lest he adjust the lamp. We forbid only when Erev Shabbos is on Yom Tov!

Poskim

Shulchan Aruch (OC 506:4): If one separates Tamei Chalah on Yom Tov, he may not burn it on Yom Tov, for we do not burn Kodshim on Yom Tov. Rather, he leaves it to burn it at night.

Bi'ur Halachah (DH Ela): The Mishnah connotes that he burns it at night. R. Yeshayah Pik asked that we hold that one may not burn Kodshim at night! It seems that the same applies to Tamei Terumah and Chalah. Also the Mishnah in Shabbos connotes that one may burn Terumah at night. It forbids burning Shemen Sereifah on (i.e. for Neros) Yom Tov. This implies that one may burn it on a Yom Chol [in a corresponding case, i.e.] at night! The Rishonim discussed this and brought the Yerushalmi (Shabbos 2:1, 15a). It says that one may burn Tamei at night according to R. Yishmael, who permits circumcision after day eight at night. We forbid Milah after day eight at night. (If so, one may not burn Tamei Terumah at night!) This requires investigation.

Note: R. Y. Glick Shlita said that the Yerushalmi means that we learn from R. Yishmael's opinion, but indeed Chachamim agree to the law.

Noda bi'Yehudah (2 OC 96): The Rambam forbids burning Tamei Kodshim on Shabbos and Yom Tov, but not at night. All the more so one may burn Tamei Terumah at night. Rabah forbids burning Shemen Sereifah on Shabbos, lest one tilt the Ner on Shabbos. If it were forbidden to burn it at night, we should likewise decree not to light during the day [even on a weekday] so it will burn at night, lest one tilt at night!