Beis Shamai: One may not take Challah and Priestly gifts to the Kohen on Yom Tov, whether they were separated before Yom Tov or on Yom Tov. Beis Hillel permit it.

Beis Shamai: It is like a Gezeirah Shavah - Challah and Priestly gifts are given to the Kohen and Terumah is given to the Kohen - just as one may not taken, so the Gifts may not be taken.

Beis Hillel respond: Terumah is different as it may not be separated on Yom Tov, but the gifts may be separated on Yom Tov.

(Gemara - Beraisa - R. Yehuda): Both Beis Shamai and Beis Hillel agree that one may take the gifts that were separated on Yom Tov and take the gifts that were separated before Yom Tov together with the gifts that were separated on Yom Tov.

The dispute is about taking the gifts that were separated on Yom Tov by themselves, that Beis Shamai prohibit it and Beis Hillel permit it.

Question against Beis Hillel: Can one not separate Terumah on Yom Tov? Surely there is a case when one made a declaration before Yom Tov verbally separating Terumah (and then actually separated it on Yom Tov)?!

Answer: They decreed against Terumah upon which one made a verbal declaration that one should not take it on Yom Tov, because of (regular) Terumah that was separated on Yom Tov.

Question: Why did the Mishnah teach a case of Chalah? (Even Terumah could be separated on Yom Tov if a declaration was made in advance!)

Answer: Chalah is preferable because it is permitted to separate on Yom Tov even without a declaration. (Note: This entry follows the Korban HaEidah's second explanation.)

It is like the Beraisa - one who kneads dough on Yom Tov separates its Chalah on Yom Tov. One who kneaded on Erev Yom Tov and forgot to separate dough, he may not move the dough and certainly may not separate the Chalah on Yom Tov.

In the second law it only spoke about if it was fully kneaded before Yom Tov - what if the water was added to the flour but it was not actually kneaded? (It seems to be permitted)

Answer (R. Shmuel, brother of R. Berachia): The case was with Tameh dough, when the Chalah is separated only at the end.

R. Yosa bei R. Bun: In truth, one should even separate Tahor Chalah at the end, but a decree was made to separate it at the beginning (when the water is added), to prevent the dough from becoming Tameh.

The Beraisa (that teaches that the dough may not be moved) refers to the Yom Tov of Pesach (as it will become Chametz), but on Shavuos and Sukos it is permitted.

R. Yosa bei R. Bun/ R. Chuna quoting R. Acha: Even on Shavuos and Sukos it may not be moved, as the Pasuk states (Shemos 12, 16), "no Melacha may be performed on them'' (and separating Chalah looks like the Melacha of fixing).