Gemara

59b: Me'ilah never applies (to something used for a Mitzvah) after the Mitzvah is finished.

Question: This is not true regarding Terumas ha'Deshen!

Pesachim 26a - Beraisa: "V'Samo" - the ashes must be placed down (and left there, one may not benefit from them).

Answer #1: Terumas ha'Deshen and Bigdei Kehunah are Shnei Kesuvim (two verses teaching the same thing, i.e. there is Me'ilah after the Mitzvah was done), so they do not teach about other laws.

Question: This is like Chachamim, who learn from "V'Hinicham Sham" that the Bigdei Lavan require Genizah. According to R. Dosa, who permits a Kohen Hedyot to use them, how can we answer?

Me'ilah 11b - Answer #2: Terumas ha'Deshen and limbs of the Se'ir ha'Mishtale'ach are Shnei Kesuvim.

Question: This is like the opinion that forbids limbs of the Se'ir ha'Mishtale'ach. According to the opinion that permits them, how can we answer?

Answer #3: Terumas ha'Deshen and Eglah Arufah are Shnei Kesuvim.

Kerisus 6a - Beraisa: "V'Orfu Sham Es ha'Eglah" - this teaches that it must be buried there.

Question: According to the opinion that Shnei Kesuvim teach to other cases, how can we answer?

Answer #4: Regarding each, a verse teaches not to learn to other cases:

It says ha'Arufah and v'Samo (but normally, there is no Me'ilah after the Mitzvah).

Me'ilah 11b - Mishnah: One may not benefit from ashes of the inner Mizbe'ach or of the Menorah; Me'ilah does not apply to them.

Question: Granted, We learn about Dishun of the outer Mizbe'ach from v'Samo. But what is the source regarding ashes of the inner Mizbe'ach?

Kidushin 54a - Beraisa - R. Meir: Me'ilah applies to worn out Bigdei Kehunah.

Avos d'R. Noson (41:12): The following were put in Genizah...Bigdei Kehunah, Bigdei Kohen Moshi'ach...

Rishonim

Rashi (Kidushin 35a, DH u'Meshani): Shenei Kesuvim do not teach about other cases at all.

Tosfos (35a, DH Ela): Shenei Kesuvim are a Mi'ut, showing that the law does not apply elsewhere.

Question: Here the Gemara (apparently) says that Me'ilah of Terumas ha'Deshen is mid'Oraisa. Me'ilah 11b connotes that it is not mid'Oraisa!

Answer #1 (Ri in Tosfos Yoma 59b, DH v'Harei): In Me'ilah it is implied that Me'ilah does not apply to Dishun of the inner Mizbe'ach and Menorah only after the Mitzvah of Dishun was done. The Gemara asks, granted, v'Samo teaches that Dishun of the outer Mizbe'ach is a Mitzvah. In fact, Me'ilah applies to those ashes even after the Mitzvah. But what is the source that Dishun of the inner Mizbe'ach is a Mitzvah?

Answer #2 (Ritzba in Tosfos ibid.): The Yerushalmi derives that Dishun of the inner Mizbe'ach is Asur b'Hana'ah mid'Oraisa. The Gemara in Me'ilah asks, granted, we know that Dishun of the outer Mizbe'ach is Asur b'Hana'ah mid'Oraisa from v'Samo. But what is the source that Dishun of the inner Mizbe'ach is Asur b'Hana'ah mid'Oraisa?

Rambam (Hilchos Klei ha'Mikdash 8:5-6): Worn out Bigdei Kohen Gadol are put in Genizah. One may not use or benefit from Bigdei Lavan that were worn on Yom Kipur. Worn out Michnesayim and Avnetim of Hedyotos are used for wicks for Simchas Beis ha'Sho'evah. Worn out Ketonos of Hedyotos are used for wicks for the Menorah.

Source (Griz Kerisus 5b DH b'Rambam (p.75)): Avos d'R. Noson says that Bigdei Kehunah and Bigdei Kohen Moshi'ach were put in Genizah. These must refer to the eight Begadim and the Bigdei Lavan of the Kohen Gadol, for worn out Bigdei Hedyot are used for wicks. This is not a special law of Bigdei Kohen Gadol. Rather, all Klei Mikdash that wore out require Genizah.

Rambam Hilchos Me'ilah (5:14): Me'ilah applies to worn out Bigdei Kehunah.

Question (Mishneh l'Melech): Me'ilah does not apply after the Mitzvah was done!

Suggestion: Perhaps we learn from Bigdei Kohen Gadol, for which Genizah is required after the Mitzvah was done!

Rejection: If so, Bigdei Hedyot should likewise require Genizah, but we learned that they are used for Simchas Beis ha'Sho'evah!

Suggestion: Perhaps they require Genizah, but nevertheless they may be used for Simchas Beis ha'Sho'evah because it is a need of the Korban!

Rejection: If so, the same should apply to used Bigdei Kohen Gadol!

Answer (Mishneh l'Melech): The Yerushalmi says that Bigdei Lavan of Yom Kipur were used inside (for the Menorah), and worn out Bigdei Kohen Hedyot were used outside (for Beis ha'Sho'evah). Surely, mid'Oraisa there is no difference. Chachamim designated the more Kodesh clothes for the Menorah.

Question: The Rambam holds that the Bavli disagrees. The Bavli learns about Bigdei Zahav from the Bigdei Lavan, and does not allow any use of Bigdei Kohen Gadol. Since it does not learn Bigdei Hedyot from Bigdei Kohen Gadol, what is the source to forbid Bigdei Hedyot after the Mitzvah?

Rambam (Hilchos Me'ilah 2:14): Me'ilah applies to ashes of the outer Mizbe'ach both before and after Terumas ha'Deshen.

Question: The Rambam does not say that Me'ilah applies to Terumas ha'Deshen itself (he holds that it does not). The Gemara says that Me'ilah applies to Terumas ha'Deshen!

Answer (Griz ibid. DH d'Havi): The Gemara means that normally, there is no Me'ilah or other Isur after the Mitzvah is done. Terumas ha'Deshen is an exception.

Question: Kidushin 54a says that Me'ilah applies to worn out Bigdei Kehunah. If so, how could they be used for Simchas Beis ha'Sho'evah, which is not a need of Korbanos?

Suggestion: Perhaps Beis Din stipulates that this will be permitted (from the beginning the Bigdei Kehunah were not totally Kodesh)!

Rejection (Tosfos Shabbos 21a DH Simchas): There is no reason for Beis Din to stipulate thusly.

Answer (Tosfos ibid.): Since it is to honor Nisuch ha'Mayim, this is considered a need of the Korban - "U'Sh'avtem Mayim b'Sason".

Question: Flutes at the time of Nisuch ha'Mayim are needed just as much for the Korban, nevertheless they do not override the Isur mid'Rabanan of instruments on Shabbos. Why were Chachamim more lenient to permit using Hekdesh, an Isur Torah?

Answer (Mishneh l'Melech Hilchos Klei ha'Mikdash 8:6): Indeed, Tosfos answers that honoring Nisuch ha'Mayim is considered a need of the Korban to justify a stipulation of Beis Din.