Doing It With Dignity
The sacrificial service in the Beit Hamikdash was performed with the dignity which is expected of a subject serving his king. The rule of thumb which the gemara here and in a number of other places applies is how one would approach an earthly ruler.
The Prophet Malachi rebuked the kohanim in the Name of G-d for showing disrespect for the sacrificial service. He accused them of conspiring to offer as sacrifices animals which were blind, lame or ill, and rhetorically challenged them.
"Please try offering it to your governor. Will he accept it and will he grant you your wish because of it?" (Malachi 1:8)
This criterion of what is fitting to offer an earthly ruler is applied in our gemara to an unusual situation. If an animal which has been designated as an olah burnt offering runs up to the top of the altar it is slaughtered there. Its carcass is flayed there and its body cut up for burning. A problem arises, however, in regard to its innards which have food wastes in them. These innards cannot be offered for burning while these wastes are in them because no one would dare bring such an excrement-laden gift to an earthly ruler. They must therefore be taken down from the altar, cleaned and only then returned for burning along with the rest of the sacrifice.
The Long Night
Jews who include in their morning prayers the recital of the chapter of "tithing the ashes" which is at the beginning of Parshat Tzav are familiar with this ritual which initiated the daily service in the Beit Hamikdash. A kohen would take a shovelful of ashes from the altar where the parts of the olah sacrifice burned through the night and place them on the floor, east of the altar ramp.
When this was done depended on circumstances. The Torah states that the olah parts can be placed upon the altar for burning "all through the night until the morning" (Vayikra 6:2). Rabbi Yochanan calls attention to this apparent redundancy and concludes that this is a subtle reference to the tithing of the ashes mentioned in the following passage. It is as if the Torah is telling us that this tithing can take place at the time when "dawn serves as the morning of the night" and even at an earlier "morning" during the night.
Since the Torah did not specify when that earlier time was our Sages fixed a timetable according to the circumstances. On an ordinary day this service would be performed around the time of the "crowing of the cock" dawn. On Yom Kippur, when this service, like all other services of the day, had to be performed by the Kohen Gadol, his heavy work schedule for the day necessitated beginning at midnight. During the Three Festivals, when Jews made their pilgrimage to the Beit Hamikdash, there were so many sacrifices to offer that by the time the cock crowed the Azara courtyard was filled with Jews anxious to have their sacrifices offered from daybreak. Since it was impossible to then perform the tithing and there was much to do in preparing the altar for this multitude of sacrifices this ritual was conducted by a kohen at the end of the first third of the night.