Do not confuse dramatic monologue here with dramatic irony. This poem is considered a dramatic monologue because the voice is but one speaker who is also a character, the Duke of Ferrara, who is addressing another (unnamed) character directly. It is not a "dialogue" because the other character does not respond.
Keep in mind that dramatic irony is when the audience knows something the readers do not know. ("Dramatic" here, refers to drama, as in a play on a stage.) Though there is an element of this due to the style of the poem, it is not the strongest category of irony represented here.
Instead, the irony used in this poem could better be classified as verbal irony and situational irony. Verbal irony is when what is said, is not necessarily what is meant. For example, in the opening lines the speaker says of the portrait of his dead wife, "I call that piece a wonder," as if complimenting the painting. He then goes on, as if further complimenting the painter (and the subject) by speaking of "countenance, the depth and passion of its earnest glance" that the painter portrays. Again, such words sound like praise. However it is obvious from the rest of the poem that the speaker is neither in awe of the painting, nor the woman portrayed. His complimentary words are used ironically, and hide bitterness and deep resentment.
In addition to the verbal irony the speaker uses, essentially, to complain about his dead wife and how she made him feel unimportant, there is an element of situational irony here as well. Situationally irony is when what happens is unexpected, or when things appear to be one thing but are actually something else. The duke claims toward the end of the poem that he would never have told his wife these things to her face ("I choose never to stoop"), as if this would have been beneath him. It is ironic because such complaints made toward a portrait after this wife has died is even lower than simply bringing them up when she was alive. Finally, by the end of the poem it seems clear that the complaints about his dead wife are a bit of a warning to a future wife. Again, the duke cannot make any of these requests in a manly nor face-to-face manner, but in a sense of weakness and passive aggression, he sends a message of behavior he does not wish to tolerate, but obviously has once, and probably will again.