What is "Korban Reishis" referring to?
Rashi and Rashbam: It refers to the Sh'tei ha'Lechem on Shavu'os, which come as Chametz. And Bikurim, which come as fruit. 1
Neither of which was brought on the Mizbe'ach (Rashbam).
Rashi writes that the only Heter is for Shtei ha'Lechem and Bikurim. Chametz is brought also with Todah, and it is not called "Reishis"!
Moshav Zekenim: Todah is not an obligation. The Torah teaches the only obligations that come from Se'or and honey.
Moshav Zekenim (7:12, citing the Ramban) #1: Lachmei Todah do not go on the Mizbe'ach for Re'ach Nicho'ach 1 . They are merely waved.
Moshav Zekenim (7:12, citing the Ramban) #2: The Torah explicitly commands Chametz in Lachmei Todah. The verse needs to teach only honey, for it is not explicit which Bikurim to bring; it taught Se'or along with honey.
Moshav Zekenim: This is difficult, for also Shtei ha'Lechem do not go on the Mizbe'ach, but they are considered l'Re'ach Nicho'ach due to the lambs with them. Also Lachmei Todah are Re'ach Nicho'ach due to the animal with them!