On kohanim who weren't eligible to eat from the sacrificial meats - beginning with a t'vul yom, one who has immersed in the mikveh for purity, but the sun has not yet set, so he isn't pure yet, for the purposes of eating from the offerings. [What's What: Mishmarot] Plus, the question of a "ba'al mum" - the degree to which even the "blemished" kohen is still a kohen in many important and privileged ways. Also, a kohen who is in acute mourning - and therefore cannot serve in the Temple at that time, but still may (or may not?) partake of the foods that were eaten by the kohanim. For example, sacrificial meats. And the Pesach offering in contrast to them... He still has to be purified by the time of eating, but there are stringencies in place with regard to these foods.