Gemara

12b - Mishnah - Beis Shamai: We do not bring Chalah or Matanos (Zero'a, Lechayim and Kevah of a Chulin Behemah) to a Kohen on Yom Tov, whether they were separated today or before Yom Tov;

Beis Hillel permit.

Beis Shamai: Chalah or Matanos are gifts to a Kohen, just like Terumah. Just like we do not bring Terumah to a Kohen on Yom Tov, we do not bring Chalah or Matanos!

We may not bring Terumah, for we may not separate it Yom Tov. You cannot learn from there to Chalah or Matanos, for we may separate these on Yom Tov!

Gemara - assumption: The Mishnah discusses Terumah, Chalah and Matanos that were separated today.

Suggestion: Our Mishnah is not like R. Yosi or R. Yehudah. It is like Acherim!

Beraisa - R. Yehudah: Beis Shamai and Beis Hillel agree that one may bring to a Kohen Matanos separated before Yom Tov with Matanos separated from animals slaughtered today. They argue about whether or not one may bring Matanos separated before Yom Tov by themselves.

Beis Shamai say, just like we may not bring Terumah (by itself), we may not bring Matanos. Beis Hillel say that we cannot learn from Terumah, for it may not be separated on Yom Tov.

R. Yosi: Beis Shamai and Beis Hillel agree that one may bring Matanos to a Kohen. They argue about bringing Terumah.

Beis Hillel say, just like we may bring Matanos, we may bring Terumah! Beis Shamai say that we cannot learn from Matanos, for they may be separated on Yom Tov.

Acherim: Beis Shamai and Beis Hillel both forbid to bring Terumah. They argue about Matanos.

Rejection: It can be R. Yehudah. The Mishnah does not discuss Matanos separated from animals that were slaughtered today. Rather, they were slaughtered before Yom Tov.

Suggestion: Our Mishnah is like R. Yehudah. It is unlike Acherim!

Rejection: It can be like Acherim. Acherim discuss animals slaughtered before Yom Tov.

Question: If so, Acherim hold like R. Yehudah!

Answer: They argue about Tefeilah (taking yesterday's Matanos, which may not be taken by themselves, together with today's Matanos).

Rav Yehudah: The Halachah follows R. Yosi.

Rava: On Yom Tov one may Molel Melilos (extract kernels from wheat in a non-skillful way) and eat them.

Question: If so, they become Tevel on Yom Tov. Our Mishnah says that one may not separate Terumah on Yom Tov (because the Chiyuv cannot come on Yom Tov)!

Answer #1: According to Rebbi they become Tevel. Our Mishnah is like R. Yosi b'Rebbi Yehudah, who exempts.

Objection: Also according to R. Yosi b'Rebbi Yehudah, if one entered them into the house to make a dough, and reconsidered for Melilos, they become Tevel on Yom Tov!

Answer #2: The Mishnah means that in most cases one may not separate Terumah on Yom Tov.

Rishonim

Rif and Rosh bring the Gemara verbatim.

Rambam (Hilchos Yom Tov 4:26): Even though one may not separate Terumah and Ma'aseros on Yom Tov, if one separated them before Yom Tov he may bring them to a Kohen on Yom Tov. All the more so one may bring Chalah and Matanos on Yom Tov.

Ran (DH Masnisin): Even though Matanos do not Tovel (forbid an animal before they are given), Chachamim permit Hafrashah lest one eat them.

Magid Mishnah: One may separate Matanos on Yom Tov because they are permitted to Zarim and they do not Tovel (so this is not Tikun).

Poskim

Shulchan Aruch (OC 506:3): If one kneaded a dough on Yom Tov, he may separate the Chalah and bring it to a Kohen.

Beis Yosef (DH ha'Lash): The Halachah follows R. Yosi, that one may even bring Terumah that was separated before Yom Tov, all the more so one may bring Chalah and Matanos. The Tur permits even if the Matanos were separated before Yom Tov. Presumably, the same applies to Chalah. Mahari Avuhav explains the Tur to be precise; only Chalah separated today may be brought. We are more lenient about Matanos because one may separate Matanos on Yom Tov even from a Behemah slaughtered before Yom Tov. Also, the Matanos are defined by the Torah, so Hafrashah has no effect. However, the Gemara permits Chalah in every case, unlike Mahari Avuhav.

Taz (5): No one would forbid to bring Chalah, even if it was separated before Yom Tov. Even Beis Shamai permit! A scribal error attributes Mahari Avuhav's words to bringing Chalah. Really, he forbids Hafrashas Chalah from a dough kneaded before Yom Tov. He explains why Matanos are more lenient.

Taz (5): Beis Shamai forbid to bring to a Kohen, even in the Yisrael's house, unless the Kohen normally eats there. Maharshal says that also according to Beis Hillel, when one may not pass through a Reshus ha'Rabim or Karmelis to bring to a Kohen, one may not give even in the Yisrael's house, unless the Kohen normally eats there. This is wrong. The Halachah follows R. Yosi, who says that Beis Hillel always permit. Hotza'ah does not apply to food on Yom Tov, therefore the Poskim do not discuss this.

Rema: On Yom Tov one may even give Chalah separated from before Yom Tov.

Mishnah Berurah (18): One may even take it through Reshus ha'Rabim on Yom Tov. One may bring it to a Kohen on Shabbos if there is an Eruv. This is only if it was baked, for raw dough is useless on Shabbos.

Kaf ha'Chayim (27): The Shulchan Aruch (YD 322:5) permits Chalah of Chutz la'Aretz to a Kohen who never saw Keri or immersed after his last Keri. The Rema says that it is the custom in our lands that Chalah is not eaten at all, since it is not eaten in Eretz Yisrael. This applies to many places of Sefard also. In such places Chalah is Muktzah. One who separates Chalah on Yom Tov may not move it after putting it down.