For the source text click/tap here: Pesachim 81
To download, click/tap here: PDF
What is the reason that the Paschal lamb must be burned before the Temple and that those who prefer to burn it elsewhere are not permitted to do so? Rabbi Yosei bar Ḥanina said: In order to embarrass them.
Presumably, the reason that most of the offering became impure is because the owners were not sufficiently careful with it. Therefore, the Sages decreed that it be burned in a public place.
This "public place" is the Bira the location of which remains uncertain and subject to to controversy between Reb Yochanan and Resh Lakish.