What is the dramatic irony in A Modest Proposal?

Let's first define irony and see if there is dramatic irony in this particular writing. 

Irony is what is known as a "figure of speech," which means that words are said, but you cannot take their meaning at face value. This means also that the intended meaning of the word is not the same as its actual meaning; rather, it potentially means the opposite.

Irony occurs in everyday life. In any given situation, irony occurs when the result is completely different from what people thought it was going to be.

In literature, we have a few types of irony:

Verbal is the type of irony explained previously, where words mean something totally different than what their original meaning entails.

If this is the type of irony you are looking for, then Jonathan Swift is your answer. "A Modest Proposal" is neither modest nor should it exist as a proposal. At the same time, it is not intended to be one; it is a macabre, long satire written by Swift in a way that we can equate with the likes of Jon Stewart, Stephen Colbert ,and all the other sarcastic comedians who say things when they really mean the opposite. In fact, the entire piece is ironic in nature.

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 boiled; and I make no doubt that it will equally serve in a fricasie, or a ragoust.

Imagine if he were serious about this!

Situational irony is what we also described previously, where something happens that nobody expected to happen because they thought that the exact opposite would result. For example, suppose that a villain is plotting the end of a superhero, only to have the entire plan fall upon him, making him his own victim. This is irony in a situation. 

Now, let's return to your question on dramatic irony. In dramatic irony, something will happen to a character that you, as the audience, are aware of, but the character is not. Think about a play where you already know that the ending will be tragic, but the characters are oblivious to it. 

Like the title says, "A Modest Proposal" is an essay that supposedly attempts to provide a solution to the "Irish problem" and the issues of poverty taking place at the time. It is a satirical speech, which does not include a plot, nor characters. For this reason, there is no dramatic irony in this work, and there could not be unless Swift had told us a tale of someone to which something happens that everyone else expected, except that very person. This being said, the type of irony in "A Modest Proposal" is mainly verbal, and not dramatic.