It is, indeed, reasonable to believe that Poe, who had "a fascination with the multiple meanings of foreign words" (Enotes) and has used several other puns in his narrative as well as much humorous irony, may have employed the literary device of pun with the name of the wine. And, one may assume that the word Cask alongside Amontillado may also suggest a play on the word casket since Fortunato--whose name is also a pun for his bad luck--is walled into a "casket" where he later becomes a bone pile much like all the others in the Montresor catacombs.
Of course, the most obvious use of pun is the play upon the word mason: the Freemason, a member of a secret society which the Catholic Church forbids its members to join, and the bricklayer. In his method of narration, Poe may have formed a sort of pun on the duello, which is the duel, or one-on-one combat, since he has his characters "duel" with words and the "dual" meaning of them, as for instance, at the end of the story when Poe retorts to Fortunato's urgings of "let us be gone" and "for the love of God," lending the phrases double meanings.
Certainly, the final paragraphs evince puns of lasciviousness and the macabre with the suggestions of sexual meanings to the phrases "Unsheathing my rapier," "ejaculated my friend" and "erected the hairs uponh my head." Moreover, the last part of the narrative suggests the enactment of an elaborate ritual that resembles what one critic calls "the profane rites of the 'Black Mass' or a parody of archetypal events." Poe's final words,
For the half of a century no mortal has disturbed them [Fortunato's bones]. In pace requiescat!
are in nature a pun upon the idea of peace. Therefore, with all the evidence of word play and suggested meaning and the puns on several proper nouns such as "Montresor," "Fortunato," and "Luchresi," it seems logical that Poe made yet another pun upon the name of the wine, "Amontillado."