Irony is continually present in A Modest Proposal, not least in the title since, if intended seriously, Swift’s proposal would be far from modest but very radical indeed. He writes that his idea would prevent “that horrid practice of women murdering their bastard children ... which would move tears and pity in the most savage and inhuman breast.”

Though the practice he describes is, in fact, horrid, the irony lies in his solution being even more appalling. This is a common feature of the essay, since there is no doubt that the real situation Swift describes is very dire and could be made the subject of a great deal of pathos without the monstrous solution he proposes.

Later in the essay, Swift say that some people “of a desponding spirit” are overly concerned about the poor “who are aged, diseased or maimed” and therefore unfit for human consumption. For this difficulty, he offers some ironic comfort, saying that no solution is necessary:

I am not in the least pain upon that matter, because it is very well known, that they are every day dying, and rotting, by cold and famine, and filth, and vermin, as fast as can be reasonably expected.

Finally, Swift ironically pretends that his solution is so attractive that he may be suspected of having some children to sell himself and protests that he personally will not profit from the scheme:

I profess in the sincerity of my heart, that I have not the least personal interest in endeavoring to promote this necessary work, having no other motive than the public good of my country, by advancing our trade, providing for infants, relieving the poor, and giving some pleasure to the rich. I have no children, by which I can propose to get a single penny; the youngest being nine years old, and my wife past child-bearing.

Irony in literature refers to stating the opposite of what thinks or knows to be true for satirical effect. One example of irony in "A Modest Proposal" is Swift's statement that "These Mothers instead of being able to work for their honest livelyhood [livelihood], are forced to employ all their time in Stroling, to beg Sustenance for their helpless Infants." The mothers Swift refers to are starving and are far from enjoying their lives while strolling. In fact, they are reduced to begging in order to survive, so Swift's reference to their leisured strolling is ironic in nature. 

Swift goes on to write that whoever can figure out how to make poor children in Ireland productive members of the country in a cheap and efficient manner "would deserve so well of the publick, as to have his Statue set up for a preserver of the Nation." This is also an ironic statement, as Swift's proposal to solve the problem of poor children is to eat them. This type of efficient and callous proposal does not deserve the recognition of the public. Instead, such a proposal is inhumane and deserves condemnation. 

Before Swift proposes his idea of eating poor children, he states that "I shall now therefore humbly propose my own thoughts, which I hope will not be lyable [liable] to the least Objection." In other words, he says that his ideas are so commonsensical and good that no one could object to them. However, his ideas are provocatively horrible and of course merit total rejection. 

A Modest Proposal by Jonathan Swift is a satirical piece on how to deal with the poor’s problems.  Basically, “Swift suggests that impoverished Irish might ease their economic troubles by selling their children as food for rich gentlemen and ladies” (enotes topic, see first link).  As the Macgill Book Review notes, “the most powerfully ironic aspect of this essay is rather obvious. The modest proposal is of course anything but modest: It is savage, frightening, perhaps even insane” (Magill Book Reviews, see second link).

Consider the opening sentence, where the speaker pretends to be concerned about the poor.

It is a melancholy object to those who walk through this great town or travel in the country, when they see the streets, the roads, and cabin doors, crowded with beggars of the female sex, followed by three, four, or six children, all in rags and importuning every passenger for an palms. (para 1)

The irony is that the speaker seems to care, but in actuality his suggestion shows that he does not see the poor as humans and really does not care about their plight, or he would not suggest selling their children to be eaten by the rich.

Another irony is that the passage was intended to make people aware of the plight of the poor and their responsibility toward it, but it is told in Stephen Colbert-style satire, pretending to be in the poor’s best interest.

I also find it ironic that this lovely tongue-in-check essay seems to also get a jab at Americans in.  The passage manages to satirize Ireland, England and American at once.

I have been assured by a very knowing American of my acquaintance in London, that a young healthy Child well Nursed is at a year Old, a most delicious, nourishing, and wholesome Food, whether Stewed, Roasted, Baked, or Boyled, and I make no doubt that it will equally serve in a Fricasie, or Ragoust. (para 9)

Apparently, Americans are heathens!  We eat our poor babies.

Read the full text here: http://andromeda.rutgers.edu/~jlynch/Texts/modest.html