A drama critic was someone who criticized an actor's performance in a dramatic role.
In 2285, following Dr. Leonard McCoy's fake death while Saavik took the Kobayashi Maru scenario test, James T. Kirk roused McCoy with the words "Physician, heal thyself." When McCoy complained that Kirk wasn't praising his death scene, Kirk replied, "I'm not a drama critic." (Star Trek II: The Wrath of Khan)