Tybalt plays a significant role as a rather minor character. His quick-temperedness at the party foreshadows the duels in Act III, scene 1. As Lord Capulet rebukes him, one can sense the rage and desire to regain his pride, seething just below the surface. Of course, he challenges Romeo to a duel, but Mercutio steps in instead. Tybalt slays him, but seemingly by accident. Romeo pursues him, and kills Tybalt in a heated battle. This leads to Romeo's banishment.
All of these are instances in which Tybalt affects the tragic nature of the play. Yet let's not forget Juliet, her parents, and the Nurse. Tybalt's death affects each of them, driving the tragic plot further. For Juliet, it forces her to reconsider her love for Romeo. Her parents drive her into marriage with Paris, thinking it is her sadness over her cousin's death that keeps her so melancholy. And the Nurse turns her back on Romeo, seeing him only as a murderer and a disdainful man. Thus, Tybalt is quite influential throughout the play.