Clarisse's most important function in the novel is to get Montag to begin to question his role as a fireman and the futuristic society's rules and norms in general. Clarisse is innocent and has not yet been victim to the societal brainwashing. She is a direct contrast to Montag's wife Mildred. Mildred buys whole heartedly into everything she sees on the parlor walls and diligently interacts with the "family", obediently takes her pills, and listens to her seashell radio.
Clarisse buys into none of it, reads, questions, and thinks for herself. She has a sense of history unlike the other people in the society. She is freespirited and freely discusses her feelings and ideas with Montag. This relationship with Clarisse, along with the incident with the woman in the house with the books, prompts Montag to question his entire existence and role in society. Clarisse's death seems to give him another push in leaving his life as a fireman.
In reading the book, I think that Montag's relationship with Clarisse was purely one of friendship. The movie alludes more to a romantic relationship, but if you've seen the movie you'll know that its a lot different than the novel.