Are there any jobs that Jews are not allowed to hold because of Jewish law?
That is a jam packed question. The answer is yes: Any job that would require you to violate a Jewish Law would be prohibited. For example, any job that requires working on the Sabbath, or some jobs dealing with non-kosher food, etc.
Another example of this is that a kohen cannot become a doctor. The reason for this is that Jewish law prohibits kohanim from coming in contact with the dead (with few exceptions). A doctor (at least in his training, if not professionally) has to regularly come in contact with the dead. This would be yet another example of a profession that is forbidden (at least to kohanim) by Jewish law.
Please let me know if there is something more specific that you had in mind.
Take care,
Rabbi Landis