A Kohen may not be a Doctor in accordance with the opinion of Rav Moshe Feinstein (אג"מ יו"ד ח"ג קנה) Rav Yoshe Ber Soloveitchik as quoted by Rav Herschel Schachter. The reason is that it is almost impossible to not come in contact or be under the same roof with a dead body.
Rav Moshe Feinstiens opinion and some interesting backround can be found in this article of the RJJ Journal:
A number of years ago, a certain kuntress (pamphlet) was published on
this question and came to the attention of Rav Shimon Schwab of
Washington Heights, who sent it to Rav Moshe Feinstein for comment.
Rav Feinstein at first refused to read it, vehemently rejecting the
very premise of the kuntress that medical training for a kohen might
be permissible. 50 His immediate reaction was that it is so obviously
forbidden that .even were the greatest rabbis in the world to try to
permit it, we would not consider their opinion. He compared it to a
passage in the Gemara (Yevamot 104) which declares that even if the
Prophet Elijah himself appears and expounds a law contrary to our
prevailing custom, we are not to listen to him. At first, Rav
Feinstein was under the impression that the author's thesis was that
since all]ews nowadays are considered to be unavoidably Tameh anyway,
there is no reason not to increase their Tl-lmeah; this premise he
dismissed as totally unaceeptable. However, finally he acceded to the
request to respond to the specifics of the kuntress. Rav Feinstein
notes that the author attempts a lenient ruling based on the argument
of pikuach nefesh - that the kohen-doctor would be able to save lives.
This reasoning, too, he rejects as "emptiness and nonsense" . One is
only obligated to heal the sick if he knows how; there is no
requirement for any person to go out and study medicine so that he
will be able to practice it! In the same way, one is not obligated to
go out and earn money to become rich so that he will have what to give
a poor man; rather, if he has money when a beggar asks, then he must
However, there have been Rabbis that permitted it. Most notably Rabbi Shlomo Goren who came up with a novel approach to mitigate the issue of Tumaas Meis.
Rav Goren has a questionable קולא About Metal Watches:
We generally assume like the רמב"ם (not like the ראב"ד) that מוסיף טומאה על טומאתו only means that a כהן who is physically touching one thing that is טמא מת may touch another thing that is טמא מת , against רבינו תם and the ראב"ד . We Also hold that מעיקר הדין a piece of metal that was in contact with טומאת מת becomes טומאת מת itself, even though for some reason the רמ"א was מקיל with regard to such metal. Based on these two ,דינים Rav Goren claimed that if a כהן wants to come into contact with a מת , all he has to do is put on a watch that is טמא מת , which the רמ"א says is מותר , and then even according to the רמב"ם one could come into contact with a מת.
(This is highly questionable. First of all, the רמב"ם may have only meant that there is no אסור דאורייתא for a כהן in contact with one מת to touch another, not that it is completely מותר. Also, the דין of מוסיף טומאה על טומאתו is learned from the word " להחלו ," which is interpreted as excluding anyone who is already מחולל ועומד . One could argue that a person is only considered מחולל ועומד when he is in contact with something that is אסור for him to come into contact with or something that is actually מטמא him. Only then is there already a חלל that coming into contact with another מת cannot worsen. It seems like somewhat of a תרתי דסתרי to paskin that one can touch the watch and then consider himself מחולל ועומד after doing so.)