Gemara

Mishnah (Parah 3:7): We are Metamei the Kohen who will burn the Parah Adumah and then he is Tovel, to rebut the Tzedukim who forbid using a Tevul Yom.

Zevachim 20b - Question: Does Tum'ah Posel Kidush (washing the hands and feet)?

21a - Answer: We Metamei the one who will burn the Parah Adumah (yet he does not need a second Kidush)!

Mishnah: They would Metamei the Kohen who will burn the Parah Adumah, then he would immerse, to disprove the Tzedukim, who said that a Tevul Yom is Pasul for the Parah Adumah.

Rejection: The Parah Adumah is different. Since a Tevul Yom is Kosher for its Avodah, Tum'ah does not Posel Kidush.

Rishonim

Tosfos (DH Lehotzi): They were not so lenient to be Metamei him with a Tum'ah mid'Oraisa. Rather, we are Metamei him with a Tum'ah mid'Rabanan.

Source (Tosfos, ibid.): The Mishnah (Parah 3:8) explains that Kohanim did Semichah on the Kohen who would burn the Parah before he burns it. The Kohanim who put their hands on him had not guarded themselves for the sake of the Parah, hence mid'Rabanan they are considered Temei'im for the Parah, and they make him Tamei mid'Rabanan.

Rebuttal (Tosfos Yom Tov, Parah 3:7): Perhaps Semichah was to honor the Kohen who burned the Parah and show fondness for him, and not to make him Tamei.

Question (Tzon Kodoshim Zevachim 21a DH Gemara): How can we settle the question of whether or not Tum'ah disqualifies Kidush from the Parah Adumah? Tosfos says that we are Metamei him with a Tum'ah mid'Rabanan! This question can be resolved.

Answer (Matzpas Eisan DH v'Ha): Perhaps the Gemara sought to prove at the least that a Tum'ah mid'Rabanan does not Posel Kidush.

v. Note: Alternatively, perhaps Tzon Kodoshim holds that the Gemara gave a better rejection, for there is no explicit source saying that they did not Metamei with a Tum'ah mid'Oraisa. Tosfos (Zevachim 17b DH Sarfah) is unsure whether or not a Tevul Yom mid'Oraisa is Pasul mid'Rabanan for Parah Adumah.

Rambam (Hilchos Parah Adumah 1:14), Rashi (Chagigah 23a DH Metam'in) and Me'iri (Yoma 2a DH uv'Inyan): They would be Metamei the Kohen who will burn the Parah with a dead Sheretz or a similar Tum'ah mid'Oraisa.

Source (Tosfos Yom Tov, ibid.): The Rambam holds that R. Yehoshua and R. Eliezer (Parah 5:4) argue about whether or not Tevilah to become Tahor from a Tum'ah mid'Rabanan, which Tzedukim do not recognize, would rebut them. He rules like R. Yehoshua, who says that it would not.

Rebuttal (Tosfos Yom Tov, ibid.) It is possible to explain the argument between R. Yehoshua and R. Eliezer differently.

SMaG (232 DH Shaninu b'Perek Gimel): Also all Kelim in which the ashes of Parah Adumah were put were Tevulei Yom.