What phrase used by the narrator is an example of hyperbole?
"Hyperbole" is a deliberate overstatement of a condition in order to emphasize its importance. "The Cask of Amontillado" begins with hyperbole: "The thousand injuries of Fortunato I had borne as I best could, but when he ventured upon insult I vowed revenge." It is improbable that the narrator was keeping track of the "injuries" Fortunato had done him, and equally improbable that that number equaled exactly one thousand. He is using overstatement to justify his passionate need for vengeance, since Fortunato has apparently moved from mere "injury" to outright "insult," an unforgivable trespass.
While this rhetorical device is often used merely for emphasis, in this tale, it should be seen (in hindsight) as an indication of the lunacy of our narrator. He is not to be trusted, but we find that out too late.