The ironic structure is a two-part structure which turns from making an assertion to undercutting that assertion, or pulling the rug out from underneath what (one had thought) had been established in the poem. The ironic structure is discussed in detail in Structure & Surprise by Christopher Bakken. Below are supplemental poems and discussion.
According to Bakken’s essay, Byron’s Don Juan exemplifies the use of this structure. For more stanzas of Byron’s Don Juan (from Canto 1), click here.
For a good example of how Byron employs the ironic structure not only in each stanza but among stanzas, read stanzas XC-XCIV (a selection culminating in a stanza included in Structure & Surprise). In these five stanzas, the young, love-sick Don Juan tries (unsuccessfully, but–at least for us readers–comically) to keep himself from fantasizing about his tutor Donna Julia by thinking about philosophy.
Here’s another poem which tries desperately (and ultimately unsuccessfully) to think about desire:
“Who will in fairest book of nature know…,” Sir Philip Sidney
And some other good examples:
“The Cathedral Is,” by John Ashbery
“We Were on the Terrace Drinking Gin and Tonics,” by John Ashbery
Armantrout’s “Dusk” plays with Whitman’s “A noiseless, patient spider…,” discussed on this blog as a poem employing the metaphor-to-meaning structure.
For more on Armantrout’s use of the ironic structure, see Hank Lazer’s “Lyricism of the Swerve: The Poetry of Rae Armantrout” (in Lyric & Spirit: Selected Essays 1996-2008 (Richmond, CA: Omnidawn, 2008), pp. 95-126; and American Women Poets in the 21st Century: Where Lyric Meets Language, edited by Claudia Rankine and Juliana Spahr (Middletown, CT: Wesleyan, 2002), pp. 27-51).
“We Real Cool,” by Gwendolyn Brooks
“My Pain Keeps Me Regular,” by Edward Thomas Herrera
“Meditation on Ruin,” by Jay Hopler
“To Help the Monkey Cross the River,” by Thomas Lux
“When the War Is Over,” by W. S. Merwin
“How I Discovered Poetry,” by Marilyn Nelson
“A World of Light,” by John Reibetanz
In a twist on the ironic structure, the following two poems are so ironic that they go beyond their obvious initial irony to become hyper-ironic, or else (or is it that is?) naturalized and not ironic.
Structure & Surprise 
[…] The above is a wonderfully funny poem created via a collaboration between Kiron Fowler and Jessica Obernuefemann, two students I met and worked with on Monday while visiting the poetry classroom of the excellent Tom McCulley at Heartland Community College in order to discuss Structure & Surprise. With its rising/falling, set-up/punch-line action, “Over a Seascape” is, of course, a fine example of the ironic structure. […]
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