daf bit: Sanhedrin 83

During the time of the temple the priests had special privileges, but also special liabilities. It was taught in a baraita: the following are liable to death at the hands of heaven: a priest who is not ritually pure or not wearing the correct garments while performing certain acts, a priest who drank wine (before serving in his role), and a priest with over-grown locks. (The baraita lists others, too.) However, if the service is performed by an uncircumcised priest, a mourner, or one who officiates while remaining seated, this is prohibited but not liable to death. (83a)

(A baraita is a teaching that is contemporary with the mishna.)

The g'mara discusses how we know each of these rulings (bringing proof texts or other citations), but does not discuss what the members of each group might have in common. I'm a little surprised that an uncircumcised priest isn't liable to death.