In the Mishna on our daf, Beit Shammai and Beit Hillel disagree about whether various sacrifices can be brought on Yom Tov. According to Beit Shammai, a korban olah, which is totally burned up, cannot be brought. A korban shelamim, however, can be brought, since parts of it will be eaten by the kohanim and by the owner, making it not only a sacrifice, but also food preparation, which is permitted on Yom Tov. Nevertheless, they forbid performing semikha on the animal. Beit Hillel permit both olot and shelamim to be brought since they are connected to the holiday, even through there is no obligation to bring them on the actual Yom Tov. They also permit semikha on both.
The holiday of Shavuot presents a particular problem for Beit Shammai; because it is a one-day holiday, the opportunities for bringing sacrifices are rather limited. According to the Mishna, if Shavuot falls out on a Friday, Beit Shammai would allow the sacrifices to be brought on Sunday, since they cannot be brought either on Yom Tov or on Shabbat (according to Beit Hillel the extra day is unnecessary, since the korbanot – sacrifices – should have been brought on Yom Tov itself). All are in agreement, however, that if Shavuot coincides with Shabbat, it will be necessary to bring the korbanot on Sunday. Nevertheless, certain things are done to ensure that everyone will recognize that Sunday is not truly a holiday (e.g. the kohen gadol wears his everyday clothing, and fasts and eulogies are permitted). This is done in order to emphasize that Shavuot does not necessarily fall out on Sunday.
The issue that the Mishna has with Shavuot and Sundays stems from the position of one of the Second Temple period sects. The Baitusim rejected the traditions of the Sages and interpreted the pasuk (verse) in Vayikra (23:15) to mean that the counting of the Omer always begins on Saturday night during Pesah, so Shavuot would always fall on Sunday. Although the Baitusim were a minority, they were apparently wealthy and influential and attempted various methods to arrange for their position to be accepted. It was therefore important to the Sages to do everything in their power to make sure that no one thought that their interpretation was correct.