Why does Poe not tell us the nature of the insult or describe any of the "thousand injuries" that the narrator suffered?
Poe's narrator Montresor talks disparagingly about Italians who deal in expensive merchandise such as gemmery with foreign millionaires. This brief passage suggests that Montresor does not consider himself an Italian and also that both he and Fortunato are business competitors dealing in art, antiques, jewelry, and other things, including fine wines. Since Fortunato is rich and well connected, while Montresor is obviously poor, it is conceivable that many of the thousand injuries in question were inflicted in business dealings. Fortunato could outbid Montresor and could afford to buy in larger quantities. When Fortunato learns that Montresor has acquired a "pipe" of Amontillado, 126 gallons, at a bargain price, he immediately thinks he might buy a whole shipload at an even better price. But he has to make sure it is the real Amontillado. He does not merely want to drink a glass of sherry or to show off his connoisseurship.
When Fortunato and Montresor reach the vaults at the bottom of the stairs, Montresor tells him, "The Montresors were a great numerous family," and Fortunato says, "I forget your arms." This in itself is somewhat insulting, since they have known each other for many years and are supposedly good friends. It is possible that the family was not a socially prominent one and did not even have a coat of arms. They were not descended from knights or royalty but from merchants or even rich peasants. And Fortunato may suspect this and be trying to embarrass Montresor. When Montresor describes his coat of arms he may be inventing it along with the Latin motto, "Nemo me impune lacessit." After all, there is no reason to assume that Montressor is telling Fortunato the truth about anything.
So that may account for many of the injuries and may be an example of the type of insult Fortunato is capable of. Another example is seen soon after when Montresor claims to be a mason. Fortunato says, "You? Impossible! A mason?" Fortunato is suggesting that Montresor would never be accepted into the Order of Freemasons but would be blackballed. In some masonic lodges it only takes one black ball to bar an applicant for membership. This is a flagrant albeit unconscious insult, and it provokes Montresor to show the trowel he has been concealing under his cloak. A trowel, of course, is an indispensible tool of stone masons.
Montresor's name shows that he is of French descent. Although his family may have lived in Italy for many generations, Fortunato may consider him an outsider, an upstart, a johnny-come-lately; and Fortunato may take opportunities to remind Montresor of his inferior social status. Montresor does not even consider himself an Italian. He criticizes Italians early in his narrative. He is not wearing a costume to participate in their big carnival. He tries to keep up appearances, but even his household servants do not respect him--possibly because he hasn't paid them lately. He is vulnerable to slurs and jibes.
Poe undoubtedly believed it is better to show rather than tell in a short story. If Montresor started to describe the thousand injuries at the beginning of his narrative, the reader might find it boring. Poe wanted to get into the action and dialogue as quickly as possible. He knew he could always allude to the injuries and insults later on. A lot of information in this masterpiece is conveyed dramatically rather than through tedious exposition.
One would think an aggrieved party, moved to murder, would be spouting off all the particular offenses incurred. But curiously, the narrator Montresor does not number the injuries because they are, to him, irrelevant. A clue to his unresponsiveness is in his family's crest, which has the Latin inscription "Nemo me impune lacessit," or "No one insults me with impunity." It's understood he has been insulted, and will therefore extract his revenge. His cool response suggests the old addage "Revenge is a dish best served cold." What Poe does with this description is characterize the cold, deliberate malfeasance of the narrator; the causes of his action to murder are unimportant, he is merely consumed with his cold passion for revenge. As the link states, one of the unique aspects of this story is that "Poe was criticized in his own time for daring to examine a crime with no apparent motive, and a murderer with no apparent remorse."