Gemara

Beraisa - Suggestion: "U'Foshat Es Begadav v'Lovash Begadim Acherim" - perhaps this is like Yom Kipur (the Kohen Gadol changes into Begadim of different Kedushah between Avodos), and a Hedyot (who has only one kind of Bigdei Kehunah) must change from Bigdei Kehunah to Chulin Begadim to take the ashes outside the Machaneh!

Rejection: "U'Foshat Es Begadav v'Lovash Begadim Acherim" equates the latter garments to the former, they are also Kodesh.

Question: If so, what do we learn from "Acherim"?

Answer: The latter Begadim are lower quality, like Tana d'vei R. Yishmael taught:

Tana d'vei R. Yishmael: The Torah teaches proper conduct. One should not mix a cup for his master in the same garments he wore while cooking food.

R. Eliezer says "Acherim v'Hotzi Es ha'Deshen" permits others, i.e. a Ba'al Mum (who may not do other Avodos) to take the ashes outside.

Beraisa - R. Yehudah: "Yilbash Al Besaro" is extra to obligate the Kohen to wear the Mitznefes and Avnet during Terumas ha'Deshen.

Likutei Halachos (DH Gali): The Torah teaches that these must be Bigdei Kehunah. This shows that Terumas ha'Deshen is an Avodah requiring all the Bigdei Kehunah.

R. Dosa says, it permits Bigdei Kohen Gadol of Yom Kipur to be used afterwards by a Kohen Hedyot.

Objections (Rebbi): The Avnet of Yom Kipur is (all linen,) unlike the Avnet of a Hedyot! Also, Begadim used for a great Kedushah may not be used for a lighter Kedushah!

Rebbi: Rather, "Yilbash" allows using deteriorated Begadim.

Zevachim 18b - Beraisa: "Vad" - Bigdei Kehunah must be linen, new, twined, the threads must be six-fold, and Chulin garments should not be worn with them.

Question (Abaye): Granted, they must be linen, but they need not be new!

Beraisa: If they are worn out, the Avodah is Kosher (b'Di'eved).

Counter-question (Rav Yosef): How do you understand 'the threads must be six-fold'? Vad connotes 'alone' (single-stranded)!

Answer to both questions (Rav Yosef): The Beraisa means, garments which are called Vad (i.e. those of a regular Kohen) should be linen, new, six-fold...

Some of these conditions are Me'akev, others are not.

Rishonim

Rambam (Hilchos Klei ha'Mikdash 8:4-5): Bigdei Kehunah must be new, nice and reach almost to the ground, like those of important people. If they were dirty or torn, Avodah done in them is Pasul. If they were deteriorated, Avodah done in them is Kosher.

Rambam (Hilchos Temidim u'Musafim 2:10,15): Terumas ha'Deshen is a Mitzvas Aseh - "V'Herim Es ha'Deshen". It is an Avodah of Kehunah. The Bigdei Kehunah used for it should be inferior to those for other Avodos - "U'Foshat...v'Lovash Begadim Acherim". The latter clothing is not Chulin, just inferior. One should not mix a cup for his master in the same garments he wore while cooking. Hotza'as ha'Deshen is not an Avodah. Nevertheless, a Ba'al Mum may not do it.

(Rashi 12b DH Es ha'Sechakim): "Yilbash" allows using deteriorated Begadim.

Source - Question: Sifri learns that the Tzibur supplies the Bigdei Lavan for Yom Kipur since it says "Bigdei ha'Bad", as opposed to 'Begadav'. Why does it say "Begadav" regarding Terumas ha'Deshen? The Tzibur supplies also these!

Answer (Kli Yakar Vayikra 1:4): This teaches that the Kohen treats them like his own clothes, i.e. he may change into inferior clothes for a dirty task.

Question (Rashi, ibid.): Some explain (like the Rambam) that "Yilbash" mandates inferior Begadim, for one should not mix a cup in the same garments used to cook. This is difficult, for "Yilbash" applies to Terumas ha'Deshen, in which only a pan's worth of ashes are taken. Inferior Begadim are only for Hotza'as ha'Deshen, in which much ash is taken and the Begadim get dirty!

Answer (Ritva, cited in Kesef Mishneh): "U'Foshat" teaches that he already changed to inferior Begadim before Terumas ha'Deshen.

Source - Ri Korkus: Reish Lakish says that R. Eliezer learns from "U'Foshat..." that a Ba'al Mum may do Terumas ha'Deshen. We do not challenge this by saying that "U'Foshat" is written regarding Hotza'as ha'Deshen! Since the Rambam holds that Hotza'as ha'Deshen is not an Avodah, surely the command to change Begadim does not apply to it, rather, to Terumas ha'Deshen.

Mishneh l'Melech (DH v'Yesh Ledakdek): One could explain that all agree that Hotza'as ha'Deshen is an Avodah since all four Bigdei Kehunah are required, and R. Eliezer learns from a verse that a Ba'al Mum may do it. It is more reasonable to say that R. Eliezer holds that it is not Avodah since a Ba'al Mum is Kosher for it; nevertheless, the verse requires the four Begadim.

Lechem Mishneh: The Rambam prefers the Perush Rashi heard because it is disgraceful to do lowly Avodah in the same garments used for dignified Avodah.

Kesef Mishneh: The Rambam did not need to say that inferior Begadim are worn for Hotza'as ha'Deshen. This is a Kal va'Chomer from Terumas ha'Deshen.

Question (Tosfos Zevachim 18b DH veha'Tanya): Here, the Beraisa does not need a Drashah to allow deteriorated Begadim (for any Avodah). Why (regarding Terumas ha'Deshen) does the Gemara in Yoma require a Drashah?

Answer (Tosfos Yoma 12b DH Yilbash): Indeed, the Gemara in Zevachim allows deteriorated Begadim because of "Yilbash".

Ramban (Vayikra 6:4): The simple meaning of the verse is like the opinion that Chulin Begadim are used for Hotza'as ha'Deshen, to avoid dirtying Bigdei Kehunah.

Nesinah l'Ger (in Ramban Tov Yerushalayim): The Ramban alludes to R. Eliezer. Just like he allows a Ba'al Mum for Hotza'as ha'Deshen, he holds that Bigdei Kehunah are not used.

Acharonim

Likutei Halachos (Yoma p.14 DH Talmud Lomar and DH Al): The inferior garments are deteriorated, or of cheap linen that is not yellowish. The garments of one who cooks absorb smoke and get stained, so they are not fit for other service. The same applies to Hotza'as ha'Deshen.