Q. Is it true that Cohanim have to stand up when saying Korbanot?

A. Mishna Berura (48: 1) quotes Magen Avrohom that one should stand when reciting korbanos, however he mentions that Sharei Teshuva, citing several Poskim disagrees. He also iterates that Pri Megadim maintains that Parshas Hatamid, that is recited aloud, should be said when standing.
Aruch Hashulchan (O.H. 1: 26) mentions that he maintains that a Cohen should stand while reciting korbonos, since if the Beis Hamikdash would be extant, he would bring the korbonos himself. However, Leviim and Yisraelim need not stand.
Nonetheless, this is not the common accepted minhag. (Machzik Brocho 48, Birkei Yosef 1: 14, Mor Uketzia ibid. et. al.).
Horav Shlomo Miller’s Shlit’a opinion is that a Cohen does not have to stand when reciting korbonos.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a.