What values does Clarisse represent? Why must she be killed or silenced?
The previous post was quite strong. I would probably expand a bit on the second question. From the social order's point of view, I think that Clarisse represents a major threat. Examine what she represents and I think that she comes into direct opposition with the prevailing political order and its expectations on the citizens. On one hand, Clarisse reveals that she likes to talk and likes to spend time with her family. This is something that cannot be controlled by the external government because it is something that transpires in the private and is unpredictable. It's fair to assume that Clarisse's conversations with her family, and with anyone for that matter, are wide ranging and initiate more thought. The fact that this cannot be contained would cause an instant threat to the Status Quo. Look at her opening question to Montag of "Are you happy?" The reader can make the clear connection that this question really starts him on his journey of doubt and alienation, another example of Clarisse's "threatening" nature. Finally, I would say that Clarisse is a threat because she is self actualized. She does not care if people perceive her as "different" because she revels in such a distinction. In the end, these individuals pose the greatest of threat to an establishment that wishes to present totality, a sense of control, and complete guarantee. She lies outside that which is quantified and because of this, she would be seen as a threat to be neutralized.
To me, Clarisse represents the values that we in our real world society would approve of. She represents what we might call more human values. Clarisse cares about people and wants them to be happy. She likes to think about things. She is really a good person and a thoughtful one.
I do not think that she really has to be killed for the society to function. After all, the society is not even who kills her -- she is not hunted down by the Hound. But for the purposes of the book, she needs to be killed because that is symbolic of how the society works. The society in the book kills off all human emotions and it kills of the desire to spend time silently thinking. By killing Clarisse, Bradbury is telling us that the values she stands for have been killed by the society of the book.