"A festival prohibition is only deferred by [a mitzvah] at its required time." What is the Scriptural source for this? Hizqia said, as it was also taught in the collection of Hizqia's academy, Scripture says: "Do not leave the [Paschal sacrifice] over to the morning, [rather burn the remainder of it in the morning.]" (Ex. 12:10) There is no need [for the verse] to repeat the word "morning." Why does it repeat the word, "morning"? To add an additional day on which the burning should occur.
Some mitzvot require us to violate Shabbat and festivals. For instance, the Torah requires that brit milah, the covenant of circumcision, take place on the eighth day of an Israelite boy's life. The eighth day is its required time, even though that day may fall on Shabbat or a festival. The same is true with regard to the mitzvah of bringing the Paschal sacrifice—our Israelite ancestors were required to slaughter their Paschal lambs and offer their blood upon the altar on the fourteenth of the Hebrew month of Nisan and eat them on the night of the fifteenth, no matter whether one of these days was Shabbat or not.
In the above passage, we see that the Torah also requires the Israelites to utterly consume the Paschal meat. Any leftovers are to be burned to ash. Exodus 12:10 states that this should be done in the "morning," seemingly the morning of the fifteenth of Nisan, the festival day of Passover. However, the repetition of the word morning leads Hizqia to conclude (in accordance with a long-standing tradition) that the burning should take place on another morning—the first non-festival morning of Passover.
The obligation to rid oneself of the Paschal leftovers is not a festival obligation. It is an obligation whose required time occurs after the conclusion of the festival day. As such, it cannot defer the festival prohibitions and must wait until the morning after the festival. The same is true of a brit milah that was delayed for the baby's health or some other reason. After the eighth day has passed, it can no longer be deferred due to Shabbat or festival prohibitions.