Irony is a state of affairs, a comment, or an event that is contrary to what you expect. We know from the beginning of the story that Montresor hates Fortunato ("The thousand injuries of Fortunato I had borne as I best could, but when he ventured upon insult I vowed revenge"). He will get his vengeance so that Fortunato knows what is happening but in such a way that Montresor himself will never be caught. In order to do both of these, he must be planning murder.
We don't know what he has in mind, precisely, when he takes Fortunato to his family dungeons in search of the cask of amontillado, but we can be certain that Fortunato will be dead by the end of it (or left to die). Thus, each time Montresor voices concern for his health (Fortunato has a wicked cough, and they are going into the cold crypts of Montresor's family home) we are struck with the incongruity of what we expect (Montresor being keen to get Fortunato to the place he plans to kill him) and what we encounter (Montresor being concerned that Fortunato will get sicker and all but ordering him to turn around and go back).