If we compare "The Dover Bitch" to Matthew Arnold's "Dover Beach," the irony lies in the comparison. While "Dover Beach" is mournful and serious, "The Dover Bitch" is straight forward and flippant. The narrator in "The Dover Bitch" is a friend or acquaintance of Matthew Arnold and he uses "The Dover Bitch" to provide commentary on Arnold and the woman Arnold speaks to in "Dover Beach."
In "Dover Beach," Arnold mourns the fact that the world is no longer united by the power of religion. "The Sea of Faith / Was once, too, at the full, and round earth's shore / Lay like the folds of a girdle furled." Arnold turns to his beloved, noting that in the absence of some unifying doctrine and/or faith, and he tells her that they should be true to one another because all they have is each other.
In "The Dover Bitch," the speaker is not nearly as serious and almost mocks Arnold:
And he said to her, 'Try to be true to me,
And I'll do the same for you, for things are bad,
All over, etc., etc.'
Then the speaker in "The Dover Bitch" mentions that he knows the girl and has even had sex with her. And while Arnold is rambling on about the loss of faith in the world, the girl is thinking about his whiskers and perfumes. The further irony, again in comparison with "Dover Beach," is how she reacts to Arnold's plea that they be true to one another. In "Dover Beach," the reader might assume that the girl is swept up in Arnold's melancholy about the loss of faith and that she is humbled that the love between the two of them (she and Arnold) is the only meaningful unifying power left in the world. But in "The Dover Bitch," she is offended that he treats her as "a sort of mournful cosmic last resort."
Then the speaker notes that the girl says something unprintable (again flippant and realistic rather than Arnold's more poetic and refined words). The speaker ends by adding that the girl is "all right" and has on occasion given her "a good time." So, there is irony in the comparison between the seriousness of "Dover Beach" and the flippancy of "The Dover Bitch." But, "The Dover Bitch" is not just a mockery of "Dover Beach." And here is even more, perhaps "meta-irony." In the first case, "The Dover Bitch" is an ironic/mock take on "Dover Beach." But it is ironic (what we might call meta-ironic) that this mocking take on "Dover Beach" is also an homage to "Dover Beach." Both poems are criticisms of life. So, while "The Dover Bitch" seems to be nothing more than a mockery of "Dover Beach," it is also, ironically, similar in theme and somewhat of an homage to "Dover Beach."