Beatty's constant quotations appear to be intentional. Here is their boss, who leads them in the burning of homes with books, quoting from the same books he is telling them to burn.
This detail seems specifically important because we know that Montag is reading and hoarding books. At first glance, it might seem that this is Beatty's way of seeing how his men will react so that he can ferret out their secrets. It is, I believe, probable that Beatty not only suspects Montag, but already knows his secrets.
Just before the firemen burn the house with the woman inside, they are playing cards at the firehouse and Montag asks Beatty about what happened to a man whose library they had burned the week before. Beatty is careful in what he says, even in the way he arranges his cards, so we might infer that this is not a casual answer he is giving Montag (which might indicate that he is trying to figure out how to "play his cards" with regard to Montag's interest in the man):
Beatty arranged his cards quietly. "Any man's insane who thinks he can fool the government and us."
Montag wonders what it would be like to be in that man's position. Beatty notes that they have no books. Montag insists—he wonders what it would be like if they did. Beatty asks: "You got some?"
If Beatty had said someone would be crazy if they tried to fool the government and had left it at that, I would probably not think twice about it. However, adding "and us" makes it personal to Beatty. In the scene when Montag sets Beatty on fire, it's almost as if it is personal again. Beatty is egging Montag on, calling him names ("snob"); alluding to Montag's foolishness by comparing him to Icarus when he flew too high to the sun; and, finally taunting Montag to recite from Shakespeare's Julius Caesar. Beatty quotes:
There is no terror, Cassius, in your threats,
For I am arm'd so strong in honesty,
That they pass by me as the idle wind
Which I respect not. (IV.iii.71-76)
I am sure that Beatty knew all along about Montag. He mentions things about Clarisse (things she said to Montag—he knew what Clarisse had said!), then that Montag was reported by his wife's friends for spouting poetry, and finally confirming that Mildred's report that Montag had books was "old news" to him. In his way, I believe that when Beatty does what no one else is allowed to do (read and remember literature), he is taunting Montag quietly, carefully and maliciously. Montag finally snaps, and even Beatty is surprised:
Beatty glanced instantly at Montag's fingers and his eyes widened the faintest bit.
Beatty tries to break Montag, but in doing so, he empowers Montag to take the step he was afraid before to take (to join other readers), and Beatty's verbal abuse costs him his life.