Clarisse's sudden death in Part One of Fahrenheit 451 is a pivotal event because of her influence on Montag. If Clarisse had not died then Montag may never have developed the "chills" and "fever" and skipped work that evening. As a result, Captain Beatty would not have visited him and realized that he was beginning to question his status as a fireman and the morality of the fireman system, more generally. Without this visit, Montag may never have shown his hidden collection of books to his wife, Mildred, and expressed his unhappiness to her.
Conversely, had Clarisse survived, her relationship with Montag would certainly continue. His association with her, however, may have alerted the authorities as to his growing sense of rebellion much sooner. With Clarisse acting as his friend and confidante, Montag may never have turned to Faber and may instead have plotted his rebellion with her.