In a sense, Montag's old life had always been a state of numbness. He was just unaware of it. When he begins talking with Clarisse and Faber, he starts to awaken from that blissful ignorance of numbness. He knows it is something he must break through.
As Montag mentally awakens, he begins to experience numbness in different ways. When he recalls his first conversation with Faber, he wants to reach out to him but his hands are numb. When Montag shows Millie the Bible, she recoils and starts talking about one of the parlour shows. Again, Montag is numb. His wife cares more about the White Clown on the show than she does for Montag himself. He feels numb because there is no real love between them. Montag is becoming aware of his numbness. With Faber, he is numb because he is afraid of confronting something new. With Millie, he feels a numbness between them.
When he reads "Dover Beach," his mouth is numb. He is trying to break through this culturally-induced behavior of numbing one's self to anything emotionally or intellectually stimulating. When Montag is running from the Hound, he is hit in the leg. The anesthesia numbs him. This is a physical numbness that is symbolic of the culturally-induced numbness in his society. The authorities dumb down the public, making them numb to new or challenging experiences and ideas. Montag becomes aware of his numbness and tries to defeat it as he tries to break through this cultural brainwashing.