Before us is Mishnah 3 in the first chapter of Maseches Chagigah. This Mishnah rests on a fundamental principle in the laws of maaser sheni: money on which maaser sheni produce was redeemed is limited in its use in two respects:
It may be used only for the purposes of eating, and not to fulfill some other obligation.
Any offering that a person is obligated to bring may not come from maaser sheni money, but only from chullin money, money that carries no sanctity.
"Olos ba'moed ba'os min ha'chullin":
The Mishnah rules that the burnt offerings brought on the festival come from chullin. The Gemara explains that the wording of the Mishnah is incomplete, and that in truth two different types of burnt offerings are under discussion:
Burnt offerings of a vow or a free will gift - which a person brings of his own accord, whether he vowed to bring an offering or set aside an animal and declared, "This shall be an offering." These are not offered on Yom Tov itself, but only on Chol HaMoed.
Burnt offerings of re'iyah - which a person is obligated to bring, and which may be offered even on Yom Tov.
Both alike may come only from chullin, and there are two reasons for this. First, a burnt offering is not eaten at all, whereas maaser sheni money is designated for the purposes of eating. Second, any obligatory offering that a person bears the responsibility to bring comes from chullin and not from maaser sheni.
"V'ha'shelamim min ha'maaser":
Against the burnt offerings the Mishnah sets the peace offerings, and the reference here is to the peace offerings of simchah. One must distinguish between two types of peace offerings:
Peace offerings of chagigah: There is a Torah obligation to bring offerings for the sake of the festival and to eat from them, and even if a person already has abundant meat from other offerings, the obligation of the chagigah peace offerings remains in force.
Peace offerings of simchah: These come only when a person has no other meat. If he has abundant chagigah peace offerings, he is under no obligation to bring peace offerings of simchah, since their entire purpose is to ensure that there will be enough meat to fulfill the mitzvah of rejoicing through eating the meat of offerings on Yom Tov.
Since the peace offerings of simchah carry no obligation of their own, and come only for the sake of fulfilling the rejoicing, one is permitted to bring them from maaser sheni money. This is the intent of the Mishnah when it says "v'ha'shelamim min ha'maaser".
The dispute of Beis Shammai and Beis Hillel regarding the first day of Pesach:
On the first day of Pesach one brings peace offerings of chagigah, and regarding these they disagreed:
"Beis Shammai omrim min ha'chullin" - because the chagigah peace offerings are an obligation, unlike the peace offerings of simchah, which carry no absolute obligation but are required only when there is no other meat. For this reason they may be brought only from chullin.
"U'Beis Hillel omrim min ha'maaser" - that they may be brought even from maaser sheni money.
The Gemara clarifies that Beis Hillel do not disagree with the fundamental principle that whatever comes as an obligation must come from chullin. Their dispute concerns only an addition: once the basic cost of the chagigah peace offering has been paid from chullin money, may a person go further and enlarge the offering with maaser sheni money. This is the intent of Beis Hillel when they say "min ha'maaser" - only the addition comes from maaser sheni, while the basic obligation is covered by chullin money.
Why did the Mishnah specify "the first day of Pesach"?
The mention of the chagigah peace offerings of the first day of Pesach comes to set them in contrast to the chagigah brought on the fourteenth, on Erev Pesach. The chagigah of the fourteenth was brought so that there would be enough meat and the korban Pesach would be eaten when one is already satisfied, and since this is not an obligation in its own right but merely a means of achieving satiety, it may be brought from maaser sheni money. In contrast to it, the Mishnah specified the first day of Yom Tov, on which the discussion concerns chagigah peace offerings that one is obligated to bring. In truth the same law applies on the first day of Sukkos and of Shavuos as well, and the Mishnah specified Pesach only in order to set it in contrast to the chagigah of Erev Pesach, which comes from maaser sheni money.
In summary: In this Mishnah we learned that maaser sheni money is designated for the purposes of eating and is not used for an obligatory offering. Burnt offerings - whether of a vow and a free will gift or of re'iyah - come from chullin, since they are not eaten and they also carry an obligation; peace offerings of simchah, which carry no obligation of their own, come from maaser sheni. Regarding the chagigah peace offerings of the first day of Pesach, Beis Shammai and Beis Hillel disagreed, and according to Beis Hillel the maaser comes only for the addition, while the essential obligation comes from chullin. The mention of the first day of Pesach comes to set it in contrast to the chagigah of the fourteenth, which comes from maaser sheni money.