I have suggested that Montresor may be spontaneously inventing his coat of arms and the motto that goes with it. He has been deceiving Fortunato about everything else, including the cask of Amontillado. Why should he be telling him the truth about his coat of arms? It is a little too appropriate for the occasion. And the Latin motto is also a little too appropriate. Fortunato is drunk, and Poe may just be having fun with him. There is a good possibility that Fortunato doesn't even understand Latin. When Montresor says the motto is "Nemo me impune lacessit," Fortunato simply says, "Good." Even a reader who does not understand Latin ought to be able to guess that it means something like "Nobody (nemo) injures me (me) with impunity (impune)." Montresor may have an entirely different coat of arms, or no coat of arms at all, and he may have an entirely different motto, or none at all. The coat of arms and motto are not necessary to convince the reader that Montresor is proud and vengeful. We are well aware of this already.
The story is set in Italy, is it not? Remember Romeo and Juliet and the family honor? Huge for this nationality is the pride in one's surname, in one's person. Certainly, the phrase "with impunity" also echoes the words of the exposition of Poe's Gothic tale, underscoring the theme of revenge, a revenge that Montesor wears upon his person as one would wear a shield. (coat of arms)
The coat of arms adds a little foreshadowing, because it tells us that revenge is in his blood. It also adds humor, because it seems silly and adds to the goofy aspect of the story. However, if you take it seriously it shows that he really is unstable.
Yes, Montressor is acting not only personally to revenge the dishonour visited upon him by Fortunato. He is also following a long family tradition in terms of not taking dishonour lightly and seeking to have his revenge on those who perpetrate such dishonour. It helps us to understand the character of Montressor a bit better.
Given that Fortunato has inflicted such "insults" upon Montressor, one could look at the coat of arms as being representative of Montressor's family. In Montressor's eyes, Fortunato did not only insult him, he insulted his family as well.
Basically, the coat of arms means "you can't mess with me and get away with it." It's a totally macho, in-your-face sort of a motto. It sets us up for what Montresor is going to do -- he's not going to let Fortunato get away with messing with him.
In Edgar Allan Poe's short story, Fortunato, the man whom Montresor plans to kill, asks Montresor about the coat of arms of the latter's family:
"I forget your arms."
"A huge human foot d'or, in a field azure; the foot crushes a serpent rampant whose fangs are imbedded in the heel."
"And the motto?"
"Nemo me impune lacessit."
The motto means "no one insults me with impunity."
These arms are obviously relevant to the story as a whole: Montresor feels that he has indeed been insulted by Fortunato. He plans to kill Fortunato (that is, he plans to crush the serpent). The detail suggesting that Fortunato's fangs have been buried in Montresor's heal can be read in at least two ways: Montresor feels that he has been injured by Fortunato; Montresor, even though successful in killing Fortunato, will be troubled by the killing for years afterward.