What arguments could be advanced to support Plato's claim that having your emotions stirred on behalf of a character in a story undermines your ability to control your own emotions?
Plato addresses this issue primarily in the Republic, but also, to a degree in Ion. His claim is based on the observation that we learn by imitation and that we form our character by habit. His theory of audience response to art is thus based on two psychological notions. The first is that imitation of models is to a large degree unthinking and unconscious. The Socratic insistence on self-knowledge is precisely intended to counteract this sort of unthinkingly imitative behavior and replace it with critical thinking. Secondly, Plato argues, in Phaedrus, inter alia, that the soul has three parts, and that habitual use of a part of the soul strengthens it. Thus using the irrational soul in response to art makes the irrational part relatively stronger and the rational relatively weaker.
Interestingly, modern advertising research (pioneered by Olgilvy) suggests that Plato was in fact correct about the unconscious nature of imitation. One of the strongest correlations with buying habits is simply how many times a person has encountered the product's name.