The Mishnah in Pesachim (82a,83a) states:
הפסח שיצא או שנטמא ישרף מיד. נטמאו הבעלים או שמתו תעובר צורתו וישרף בט״ז. ר׳ יוחנן בן ברוקה אומר אף זה ישרף מיד לפי שאין לו אוכלין. העצמות והגידין והנותר ישרפו בט״ז. חל ט״ז להיות בשבת ישרפו בי״ז לפי שאינן דוחין לא את השבת ולא את היום טוב.
A Pesach which left [Yerushalayim] or became tamei is immediately burned. If the owners became tamei or died, we leave the Karban over and burn it on the 16th. R’ Yochanan Ben Beroka said, even in this case it’s burned immediately, since there are no eaters. Bones, sinews, and leftover Karbanos are burned on the 16th. If the 16th fell on Shabbos, we burn it on the 17th, since it doesn’t push off Shabbos nor Yom Tov.
When I learned these Mishnayos with my 10-year-old Chavrusa, he asked me a very simple question, which I was kind of embarrassed that I didn’t come up with on my own.
The Halacha follows Rav (Gemara on 6a) that one who found Chametz on Pesach should cover it and burn it on Chol HaMoed (OC 446:1). Thus, he posed a simple Kal v’Chomer: if Chametz, which has a prohibition against owning it, we still wait to burn it, a Tamei Karban Pesach, which does not have a prohibition against owning it, is it not logical that we should wait to burn it? So why in the beginning of the Mishnah does the Tanna Kamma say that we burn it immediately?
Then we read a bit further and got to the end of the following Mishnah, which said that it doesn’t push off Shabbos and Yom Tov, further strengthening his question. I checked the Gemara, and indeed, 84b-85a gives not one but four different sources that this is the case (Chizkiya’s source that the passuk has an extra “ad boker,” Abaye’s source of “olas Shabbas biShabbato,” Rava’s comparison to a Bris Milah that was pushed off, and Rav Ashi’s understanding that this is an aseh versus an aseh and a lav). Even if his Kal v’Chomer fell apart, he has a Mishnah, defended by several pesukim, that support his question!
I originally thought to answer that Nosar, the subject of the end of the Mishnah, is different than a Tamei or Passul, the subject of the beginning, but then I saw that these same expositions appear in Shabbos 24b-25a regarding the Mishnah of אין מדליקין שמן שרפה ביו״ט, we don’t light on Yom Tov with Shemen Sereifah (Terumah oil that’s Tamei and must be burned). Since the same pesukim are applied by Shemen Sereifah and Nosar, there’s clearly no distinction between Tamei and Nosar. Why, then does the Tanna Kamma in our Mishnah in Pesachim learn that we burn a Tamei or Passul Karban Pesach immediately?