"The Cask of Amontillado" is a tale of extreme revenge in which a man dies in one of the most frightening ways that can be imagined. Given this premise, the story is surprisingly lacking in overt conflict. Fortunato goes willingly towards his death and does not protest or struggle until it is too late. Nor does there appear to be any internal conflict in Montresor, who does not question that the "thousand injuries of Fortunato" have merited this grisly death.
Given that the conflict between Montresor and Fortunato is mysterious and may be altogether the product of Montresor's imagination (since Fortunato seems blithely unaware of it), Montresor has to keep engineering conflicts throughout the story to override any doubts or suspicions Fortunato might otherwise have. This he does very skillfully, knowing Fortunato's weaknesses: his stubborn pride and his vanity about his knowledge of wine.
The first conflict Montresor sets up in this way involves Luchesi, a rival connoisseur of wine. Fortunato angrily dismisses the idea that Luchesi has a "critical turn" where wines are concerned and insists on accompanying Montresor to his vault. The next time Montresor sets up another conflict of this kind, he even adds Luchesi as a grace note at the end, to make quite sure that Fortunato will oppose him:
"Come," I said, with decision, "we will go back; your health is precious. You are rich, respected, admired, beloved; you are happy, as once I was. You are a man to be missed. For me it is no matter. We will go back; you will be ill, and I cannot be responsible. Besides, there is Luchesi—"
Montresor, therefore, creates conflicts along the way as they go through the vaults, to make Fortunato the most powerful advocate for his own destruction.