Dear Rabbi Kornfeld:
At the bottom of daf 23b, the germara is refuting Rebbi Yehoshua ben Levi, who says we learn the isur hana'ah on Chametz and Shor ha'Niskal from the Pasuk in Parshas Tzav; "Kol Chatas Asher Yuva mi'Damah...ba'Esh Tisaref". The gemara says we already know that halacha from the pasuk in Parshas Shemini; "ve'Hineh Soraf..."; and Rashi says here (d'h "Im eino inyan") that this refers to the Chatas of Nachshon (brought on the day of the dedication of the Mishkan) as one of the 3 goats sacrificed on Rosh Chodesh Nisan. It was amazing therefore, to see Rashi on that pasuk in Chumash say that it was the korban of Rosh Chodesh, and Tosefos (d'h "dich'siv") also says here on the daf that it was indeed the korban of Rosh Chodesh.
How can we explain this Rashi here on the daf that says it was the Korban of Nachshon?
Your point is alluded to in a remark in the margin of the Vilna Shas, which adds that Rashi himself explains later (82b, DH l'Kach Ne'emar) that the Korban refers to the goat offering of Rosh Chodesh. Rashi's words are indeed very difficult to understand, because the Gemara in Zevachim (101b) cites a Beraisa which states explicitly that the Korban of Rosh Chodesh was burned, as the Tosfos Rabeinu Peretz and Tosfos ha'Rosh point out. I did not find anyone who gives any resolution for Rashi.
However, it is possible to find support for Rashi's explanation here from the Beraisa there, which describes how Moshe asked Aharon two questions: (a) Why was the Rosh Chodesh offering burned and (b) why were the other offerings of that day not burned. From there it is implied that for the same reason for which it is proper to burn the goat offering of Rosh Chodesh, the goat offering of Nachshon is also fit to be burned. If so, Rashi's words in our Sugya are correct; it can be learned from the story of Se'ir Rosh Chodesh that Se'ir Nachshon should also be burned.
Nevertheless, we are left with the question why Rashi of found it necessary to explain the Gemara here like that; even if the verse, "ve'Hineh Soraf," refers only to the Korban of Rosh Chodesh, we would have learned from there that any Korban whose blood is brought within the Heichal has a Din of Sereifah.
Perhaps it could be that if the verse is referring to the goat offered on Rosh Chodesh, we would not be able to learn from that verse that the Chatas of an individual is burned when its blood is brought in the Kodshei Kodshim; the verse would only apply to public Korbanos. For this reason Rashi mentions that it refers not only to the goat of Rosh Chodesh, but also to that of Nachshon, which was a Chatas of an individual, and thus we have a source that the Chatas of an individual is also burned.