In "Fahrenheit 451," blood symbolizes the primal, repressed part of humanity. For example, Montag's revolutionary thoughts and actions, especially where it concerns illegal and hidden knowledge, are accompanied by an awareness of his blood, such as when it wells, flows and pumps through his heart.
Blood symbolizes both that which is repressed in humans and the human primal self. Everything that Montag does or thinks that is illegal or taboo in society is followed by some sort of mention of blood, whether it is a pounding heart or a rushing of blood in his veins. The primal, fundamental importance of blood is seen when Mildred, Montag's wife is revived. Her poisoned and corrupted blood, that part of humanity that the society is trying to destroy, is easily replaced by something artificial, just like the society is trying to replace humanity with something artificial and fake.
This symbol of blood also plays into larger themes in the book. According to Reference.com, one of these themes is censorship so extreme that it destroys thought, resulting in apathy and repression of humanity. In fact, one 1967 edition of "Fahrenheit 451" had censored out a few of the words Ray Bradbury had originally added into the text to make it more student-friendly.