The Sefer haChinuch (603) opines that women are exempt from the mitzvah of zechiras Amalek because they are not participants in war and do not to go out to battle; since they would not wage war against Amalek, they are exempt from the mitzvah of zechira as well. The assumption (which the Minchas Chinuch questions) of the Chinuch is that the mitzvah of zecher and the mitzvah of mechiyas Amalek go hand in hand.
R’ Wahrman in his Sheiris Yosef (vol 7) refers to a fascinating teshuvah of the Binyan Shlomo (#57) who suggests that kohanim did not go out to battle either, as they could not risk becoming tamei meis; therefore, they too were exempt from the mitzvah of waging war with Amalek. The Binyan Shlomo writes that if this is true, kohanim should be exempt from ta’anis Esther, as according to Rabineu Tam the ta'anis is a commemoration of the fast undertaken on 13Adar in preparation for the war against Amalek. Since kohanim did not participate in battle, they would not have participated in the fast. (I am not sure why this follows – perhaps there is a chiyuv to join with the tzibur in doing teshuvah and fasting to ensure success in battle even if one does not participate directly in the fighting.) It follows as well that according to the logic of the Chinuch, kohanim should be exempt from the mitzvah of hearing parshas zachor.
The gemara's question of whether a kohein can take a yefas to’ar (Kiddushin 23) implies that kohanim did in fact go to battle, contrary to the Binyan Shlomo, but there does seem to be some basis in Rishonim for his view. In the context of debating whether a kohein who gives a get al tnai may remain together with his wife, the Mordechai (Gittin 432) attempts to bring proof from the fact that all who went out to war with King David wrote gittin (al tnai, lest they be lost in war), presumably including kohanim, who returned afterwards to their wives. The Mordechai is not convinced of the proof, as one could argue that kohanim never went out to war to begin with (he raises the sugya in Kid as an issue, but how he deals with it is unclear to me.) Anyone in tune with the current debate over whether yeshiva students should server in Tzahal has probably heard the oft-cited Rambam (end of Hil Shemita) that anyone who wants can take on the role of a levi as teacher to Torah and become exempt from the burden of going out to war. Whether the Rambam’s view can serve as a basis for yeshiva student levi-wanna-be’s being exempted may be debatable, but it seems clear that a true kohen or levi could indeed stay home. Or maybe it’s not so clear: the Rambam himself (Hil Melachim ch 7) refers to a kohein married to a gerusha as being exempted from battle because of the aveira he has done, implying that a kohein otherwise would in fact have to join the fight.