It is forbidden to separate Terumah on Shabbos and Yom Tov, but one may separate Challah from a loaf being baked on Yom Tov. Our Mishnah discusses giving the separated Challah to a kohen, since once you’re giving you might separate what’s forbidden.
Mishnah:
Beis Shammai: One cannot bring challah and matnos kehunah [certain parts of a slaughtered animal which must be given to a kohen] to a kohen on Yom Tov [Tosfos explain that it’s not because of the prohibition against carrying; even if the kohen comes to pick it up, it’s forbidden] whether it was separated before Yom Tov or on Yom Tov. Beis Hillel permit it.
Beis Shammai: There’s a comparison to be made here; challah and matanos are given to the kohen just as terumah is given to the kohen. Since you agree that terumah cannot be brought on Yom Tov, you should agree that the same goes for matanos.
Beis Hillel: No. One has no right to separate terumah on Yom Tov, but he has the right to separate matanos on Yom Tov.
Gemara:
It enters our mind: When Beis Shammai said “whether it was separated before Yom Tov or on Yom Tov”, it means “whether the animal was slaughtered before Yom Tov or on Yom Tov”. The Mishnah is not according to Rabbi Yosi or Rabbi Yehuda, but according to Acherim.
Beraisa: Rabbi Yehuda: Beis Shammai and Beis Hillel agree that matanos separated before Yom Tov may be taken to the kohen along with other matanos separated on Yom Tov, their debate is only regarding bringing matanos separated before Yom Tov on their own. 
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