Based on Moses’ instructions, Aaron and his sons ate their portions of the sacrifices that were prescribed for that day: the installation grain-offering, the installation sin-offering, the installation peace-offerings, and Nachshon’s sin- and grain-offerings. There was, however, another offering of superior holiness that they would normally have been required to eat: the goat brought as the sin-offering accompanying the additional ascent-offerings for the first of the month.
However, unlike the other offerings, this one was not an exceptional offering prescribed specifically for that day, but a regular offering that was to be brought every month. Since Moses, when he informed Aaron and his sons that God had instructed them to eat their portions of the sacrifices even though they were mourning their close relatives, had only mentioned the two grain-offerings that were prescribed solely for that day, they assumed that this directive only applied to such sacrifices and not to regular sacrifices that were prescribed for all time. The regular rule, they reasoned, still applied to regular sacrifices, meaning that they were forbidden to eat the meat of this goat. (Waiting until the next day to eat it—when they would no longer be in mourning—was not an option, since sacrifices of superior holiness may not be eaten beyond the day they are offered up and the following night.)
Therefore, instead of eating their portion of the sin-offering goat brought on the first of the month, they burned it up, as should be done with any sacrifice that cannot be eaten within its prescribed time.
