There are some korbanot that must be burned entirely after their blood is sprinkled on the altar and their innards are sacrificed. Thus, many of the Yom Kippur sacrifices, as well as some of the public guilt offerings (for example, those brought by the kohen gadol, and those brought by the Great Sanhedrin that erred and caused the majority of the community to sin) were taken to the beit ha-deshen – the place of the ashes – to be burned (see Vayikra 4:12). If, however, a korban is burned because it must be destroyed, e.g. some error or blemish kept it from being brought as a sacrifice, the Mishnah on today’s daf (=page) teaches that it is not taken to the beit ha-deshen, rather it is burned in the beit ha-birah.
What were these places where the sacrifices were burned?
Rav Nahman quotes Rabbah bar Avuha as teaching that there were three places in the environs of the Temple that served as repositories for ashes.
1. There was a large beit deshen in the Temple courtyard where they burned the holiest sacrifices and the innards of lesser sacrifices that had become disqualified.
2. There was a second beit deshen on the Temple Mount where they burned the animals that were supposed to be burned that had become disqualified after sprinkling the blood.
3. The third beit deshen was for those korbanot that were done properly and had to be burned according to their basic requirements. This beit deshen was outside of the three camps (the inner camp of the Tabernacle, the middle camp of the tribe of Levi and the outer camp of Israelites) in the desert. When the Temple stood, this was outside the walls of the city of Jerusalem.
As far as defining the term beit ha-birah, Rabbah bar bar Hannah taught that it was a particular spot on the Temple Mount, while according to Rabbi Yohanan the entire Temple was called Birah (see Sefer Divrei HaYamim (I 29:19).