Verbal irony is the intentional use of words to mean something different from what the character actually says. Throughout the short story, Montresor continually uses verbal irony during his conversations with Fortunato in order to encourage Fortunato to follow him down into his family's catacombs. 

• "My friend, no; I will not impose upon your good nature" (Poe, 2).
• When Fortunato makes a toast, Montresor responds by saying, "And I to your long life" (Poe, 3).

Situational irony is a discrepancy between what is expected to happen and what actually happens. The short story takes place during carnival season, which is generally happy, fun-filled time. However, Montresor plans and executes a horrific murder during the festive, joyful time.

Dramatic irony is when the audience is aware of something that the characters are not. Throughout the short story, the audience is aware of Montresor’s evil intentions to harm Fortunato the entire time. However, Fortunato is unaware that he is following Montresor to his death. 

The Cask of Amontillado, perhaps the most famous of Edgar Allan Poe's short stories, is not only the horrific tale of a man immured alive, but also a tour de force of the use of irony. Poe provides the raison d'etre for this when he writes from the outset that "it must be understood that neither by word nor deed had I given Fortunato cause to doubt my good will." From the beginning to the end of this tale of deception, therefore, various kinds of irony are on display.

Verbal irony - saying one thing but meaning the opposite - appears in the greeting Montresor has for the doomed Fortunato: "you are luckily met"; in Montresor's feigned concern for his friend's hacking cough in his damp catacombs: "Your health is precious...You are a man to be missed"; and in Fortunato's reply: "The cough is a mere nothing; it will not kill me. I will not die of a cough."   

Situational irony - when events turn out the opposite of what ought to have been expected - appears when Montresor, on the night of carnival, orders his servants not to leave in his absence, thereby ensuring they would do the opposite; when the non-existent cask of amontillado turns instead into the 'casket' for the unfortunate conoisseur; and when premeditated murder remains unpunished even after fifty years.  

Dramatic irony - when readers know more than the characters - is present in the very name of Montresor's 'enemy', a most unlucky man; is present in Fortunato's doomed fool costume of "tight-fitting parti-striped dress, and [...] head...surmounted by the conical cap and bells"; and is present in the trowel Montresor reveals, sign of Masonic brotherhood to Fortunato, but tool of immurement to Montresor. 

Here are some examples of irony from "The Cask of Amontillado" that you have requested.

VERBAL IRONY.  Fortunato claims that "I will not die of a cough." Montressor answers, "True. True." This is ironic because Montressor already knows how Fortunato will meet his end.

SITUATIONAL IRONY.  Fortunato is dressed "motley." He wears the costume of a court jester--a fool. He will prove just how foolish he is by following Montressor to the exact location he has planned for Fortunato's final resting place.

DRAMATIC IRONY.  When Fortunato asks Montressor if he is a member of the Free-Masons (a secretive fraternal organization), Montressor responds that he is. Fortunato asks for a secret sign that all Masons would recognize. Montressor produces a trowel--a common tool used by masonry workers. Fortunato merely scoffs at his companion, but he should have wondered why Montressor was carrying such an object.