Rav Shimon Schwab apparently asked Rav Moshe Feinstein about a pamphlet published in the mid-1970's that permitted kohanim to become doctors. Rav Moshe rejected the two major premises for this היתר:
1) The premise that since we are all טמאים טומאת מת anyway, therefore there is no harm done by kohanim going to medical school (and thus working with cadavers). Rav Moshe vehemently rejects this argument, based largely on his seeing this as a major חידוש that thus violates several Talmudic statements that rail against making rulings that go against accepted practice.
2) The second premise was that it is permitted for a kohein to practice medicine since פיקוח נפשות is involved. Rav Moshe rejects this argument as being a significant and misdirected expansion of the permissive side of פיקוח נפש. While it is true that one may violate certain laws in order to save someone else's life, that only applies when it is within one's power and ability to save the life right now. However, to go to medical school in order to increase one's ability to possibly save another's life (which there is no guarantee that one will be able to do anyway), does not outweigh the Torah-based prohibition against kohanim becoming טמאי מת.