In "Harrison Bergeron," it is not simply the case that Kurt Vonnegut employs heavy use of irony but, moreover, he utilizes the different kinds of irony to searing effect. Generally speaking, the use of irony is divided into three subcategories: situational irony, dramatic irony, and verbal irony. Vonnegut uses all three of these forms and occasionally will even employ more than one of them simultaneously.
An example of this can be seen in the short story's opening paragraph, which combines verbal irony and situational irony:
The year was 2081, and everybody was finally equal. They weren't only equal before God and the law. They were equal every which way.
To summarize, verbal irony tends to rely on sarcasm, where the literal meaning of the words are in opposition with the intended meaning behind it. In this case, we have the narrator allegedly praising the attainment of genuine equality, even as he is (in truth) ridiculing the very ideal of equality in this sense. As the paragraph continues from here, and as we begin to learn more about what this equality entails, Vonnegut begins to introduce the situational irony that will shape his story in general.
When we imagine equality, we tend to envision it as an ideal, imagining that a future that successfully achieves this ideal would be a more just society than the one we currently live in. But Vonnegut ruthlessly upends those expectations, exploring a world where equality is imposed, and all forms of genuine self-actualization and excellence are ruthlessly suppressed through the use of force. In the process, his ironic tone begins to approach a level of absurdity: with their handicaps in effect, for example, the ballet dancers cannot dance (even though that is their occupation), thus defeating the very purpose of having televised ballet to begin with. Vonnegut's picture of an equal society, thus, takes on a degree of madness: it is entirely irrational, which is precisely the point he is making regarding the very notion of equality to begin with.
Finally, there is dramatic irony, where the audience is aware of story-relevant details even as the characters are not. An example of this can be found in the scene where Hazel suggests that George relieve some of his handicap and relax while in the safety of his house. Here we see George alluding to what he refers to as "the dark ages," a time "with everybody competing against everybody else." Ultimately, in this scene, they are speaking disparagingly about the society of the real world, viewing it as a mad, unjust dystopia, even as readers would view the same of the fictional society in which they reside.
All of these uses of irony add up towards a brutally satirical picture of what equality really means and the methods that might be required to impose it. In so doing, Vonnegut is trying to hold up to criticism and ridicule the very utopian ideals that lie behind this notion of equality altogether. For Vonnegut, it seems, this form of equality amounts to a denial of human achievement and can only result in widespread misery.