As Fortunato and Montresor are descending into Montresor's vaults, they enjoy a draught of Medoc to ward off the cold. Fortunato says, "I drink..to the buried that repose around us." Montresor responds with his toast: "And I to your long life."

Montresor's toast is ironic because it is his intent to bury Fortunato in his vaults.  They continue to descend lower and lower into the Montresor family catacombs until they reach a crypt piled on three sides with human remains. In the fourth wall, there is another internal crypt, and Montresor lures Fortunato into this crypt with the promise of showing him the cask of Amontillado. Then, he fetters Fortunato to the granite and builds a wall to enclose the crypt. There, Fortunato will be buried. Therefore, while Montresor is toasting his friend's long life, he plans to soon have his friend meet an untimely demise in his crypt. 

What is ironic about Montresor's concern for Fortunato's health in "The Cask of Amontillado"?

What is ironic, or deliberately contrary to what the Fortunato expects, about Montresor's concern for his health is that Montresor's real intent is to cause harm to the man.

Another interesting aspect is that Montresor's verbal irony has not only the intent of deceiving Fortunato that he is worried about his cough, but it also has a subversive motive behind it, as well. For, in addition to feigning concern for Fortunato, Montresor repeats the opposite of what he means in order to spur Fortunato to move farther and farther into the catacombs more recklessly and irrationally. Evidently, Montresor has discovered what Mark Twain calls "a great law of human action; namely,

...that in order to make a man or a boy covet a thing, it is only necessary to make the thing difficult to attain. [The Adventures of Tom Sawyer]

Thus, with the ironic concern and urging for them to turn back, Montresor makes "the thing (the Amontillado) difficult to attain," so that Fortunato covets it all the more, and in his cupidity loses his reason to the point that when Montresor fetters him to a wall, Fortunato is "too much astounded to resist." 

What is ironic in Edgar Allan Poe's "The Cask of Amontillado"?

Poe's masterful short story has several instances of irony in its narrative:

• The victim's name is Fortunato.
• Early in the story, the narrator refers to Fortunato as "my friend."
• Fortunato's costume for the carnival is that of a clown, "motley."
• In light of what happens to him, it is ironic that Fortunato insists upon going to the vaults of Montresor.
• With verbal irony, Montresor tells Fortunato that they should go back because of Fortunato's cough 
• Fortunato tells Montresor to not worry; "I shall not die of a cough." (He will die, but of something else.)
• Fortunato makes a toast "to the buried that repose around us. (He will soon be one of them.)
• Montresor's coat of arms has the motto of the royal arms of Scotland: "No one can attack me without being punished." (Fortunato will soon be "punished.")
• Fortunato makes the sign for the Masons, asking if Montresor is one. Montresor waves a trowel around and declares that he is a mason, making a pun on the word's meanings for the Freemasons and the brick layer. The trowel, ironically, will be used to entomb Fortunato.
• As they venture deeper and deeper into the catacombs, Montresor says, "herein is the Amontillado. As for Luchesi--" and Fortunato interrupts, "He is an ignoramus." However, it appears that Fortunato is the fool as in the next moments Montresor has him fettered to the granite wall.
• Montresor, having prevented Fortunato from going anywhere by padlocking a chain around him, then points to the niter seeping from the wall, saying with verbal irony,

Once more let me implore you to return. No? Then, I must positively leave you. But I must first render you all the little attentions in my power.

• After he finishes walling in his victim, whom he has tricked through manipulation of his foolish pride, Montresor narrates that he can barely recognize the voice of "the noble Fortunato" as though he has some respect for his victim.
• Fortunato pleads, "For the love of God, Montresor," and Montresor replies alike with an ironic twist upon the words, "'Yes,' I said. 'for the love of God.'"
• For some reason, Montresor calls to Fortunato, but when he does not answer, Montresor comments ironically that his "heart grew sick--on account of the dampness of the catacombs," not because he wished to further torture his victim. 
• Commenting that for over fifty years, no one has disturbed the "old rampart of bones," Montresor says, "In pace requiescat"  with verbal irony as he wishes the bones to remain untouched, rather than wishing Fortunato's soul to rest in peace.