Chapter 4, Mishnah 4. This Mishnah comes to clarify what was done with the leftovers of the Terumah of the shekels. The three baskets, the three containers, from which the Terumah was taken, had their funds spent on the needs of the offerings; but at the end of that period, once the second Terumah had been taken, what was done with the remainder of the first Terumah? That is the question of the Mishnah.
The opinion of the first Tanna:
From the surplus of the Terumah they made a sheet of gold, a plating for the Holy of Holies. The walls and the floor of the Holy of Holies were covered with gold: they beat the gold into an extremely thin sheet, and then attached it to the walls and the floor.
The view of Rabbi Yishmael:
Rabbi Yishmael disagrees, and his words here rest on what he said in the previous Mishnah. There the law of the remnants of the Chamber was discussed, the money left in the Chamber that is never taken as Terumah. Rabbi Yishmael said that these funds are taken and invested in wine, oil and flour, which are then sold to those who need them for bringing offerings in the Beis HaMikdash, and the profits go to the Temple treasury. Rabbi Akiva disagreed and said: one does not do business with consecrated funds.
And since Rabbi Yishmael holds that one may do business with consecrated funds, in our Mishnah he distinguishes between two kinds of surplus:
"Mosar HaPeiros" - "peiros" literally means fruit, but the intention here is the profits from that very commercial activity he proposed in the previous Mishnah. These go to the "dessert of the altar", the burnt offerings brought on the altar at a time when there is no other service upon it.
"Mosar HaTerumah" - that same surplus of the Terumah which the first Tanna applied to the gold plating serves, according to Rabbi Yishmael, for making vessels in the Beis HaMikdash.
The view of Rabbi Akiva:
Rabbi Akiva, who disagreed in the previous Mishnah and held that one does not do this with consecrated funds, does not recognize the concept of "peiros" at all. Therefore the money with which the burnt offerings for the "dessert of the altar" are purchased, for the time when nothing else is being offered on the altar, he draws from a different source: "Mosar HaTerumah". In this he disagrees even with the first Tanna, who held that the surplus of the Terumah was used for the gold sheeting of the Holy of Holies.
For the service vessels Rabbi Akiva brings another type of surplus funds: "Mosar Nesachim" - the leftovers from the libations, from the wine that was poured on the altar, and included in this are also the meal offerings that were brought with the communal offerings. Even though there was no intention to do business with consecrated funds, at times the Temple treasury did profit: those who supplied the wine, oil and flour to the Beis HaMikdash committed themselves to a fixed price. If the value rose and the items became more expensive, they were obligated to supply at the cheap price; and if the price fell, they were obligated to supply the larger quantity. In this way a surplus was sometimes created in the quantity that the Temple treasury received and did not need, and this surplus is called "Mosar Nesachim". From it, according to Rabbi Akiva, the service vessels are purchased, and not from the surplus of the Terumah, as Rabbi Yishmael said.
The view of Rabbi Chanina the Deputy of the Kohanim:
Rabbi Chanina the Deputy of the Kohanim reverses the arrangement: "Mosar Nesachim", that same surplus value created by the fluctuation in prices, which Rabbi Akiva introduced into the discussion, is used for the "dessert of the altar", to provide offerings for the time when the altar has nothing else to offer; while "Mosar HaTerumah", from the Terumah of the Chamber, is designated for the service vessels, as Rabbi Yishmael said.
"Zeh V'Zeh Lo Hayu Modim B'Peiros":
The Mishnah concludes by establishing that both Rabbi Akiva and Rabbi Chanina the Deputy of the Kohanim did not concede regarding "peiros", they did not agree with Rabbi Yishmael that such a thing exists. "Peiros", as stated, is what Rabbi Yishmael described in the previous Mishnah: conducting business with consecrated funds in a manner that involves a planned profit for the Temple treasury, through selling the wine, oil and fine flour for the meal offerings and libations in the Beis HaMikdash. This practice they did not accept.
In summary: The Tannaim disagreed about the designation of the surplus funds: according to the first Tanna, the surplus of the Terumah was designated for the gold sheeting of the Holy of Holies; according to Rabbi Yishmael, the surplus of the profits was for the "dessert of the altar" and the surplus of the Terumah for the service vessels; according to Rabbi Akiva, the surplus of the Terumah was for the "dessert of the altar" and the surplus of the libations for the service vessels; and according to Rabbi Chanina the Deputy of the Kohanim, the surplus of the libations was for the "dessert of the altar" and the surplus of the Terumah for the service vessels. And both Rabbi Akiva and Rabbi Chanina alike do not concede at all to the existence of "peiros".