Ultimately, "Harrison Bergeron" is a criticism of egalitarianism (as well as society's tendency to advance social progress through government action). Vonnegut advances this criticism by creating a dystopia in which everyone has been made equal through the use of coercive force. In characterizing this dystopia, Vonnegut makes heavy use of irony and exaggeration, in order to convey the irrationality and madness of this world he is depicting.
First, note that "Harrison Bergeron" is a dystopian story: it provides a picture of the future, set in 2081, in which people have been made equal through constitutional amendment. This has been achieved through the use of handicaps and is backed by threat of force. As is often the case with dystopian stories, Vonnegut uses this vision of the future to provide a warning to his own society, depicting these egalitarian ideals as potentially dangerous and dehumanizing.
Probably the one literary device that ties this story together more than any other is irony. At the heart of irony, you will always find a tension. Note that irony can generally be divided into three kinds: verbal, situational, and dramatic. What is unique about "Harrison Bergeron" is that it uses all three.
In situational irony, the tension lies between expectations and outcomes. In this case, Vonnegut's entire dystopian vision of the future is an example of situational irony: we would expect that a world where all people are equal would result in a more utopian picture, but Vonnegut subverts those expectations.
Verbal irony, on the other hand, involves a tension within the language of the text, by which its literal meaning conflicts with its intended message. You can see an example of this in the story's opening paragraph, beginning from the story's first sentence: "It was the year 2081, and everybody was finally equal." Consider these words, taken independently of any context, and now apply them within the context of Vonnegut's dystopia: that sense of contradiction is an example of verbal irony in practice.
Finally, in dramatic irony, readers are aware of information that the characters within the story do not have. In this case, readers are aware of this society's irrationality and madness. Vonnegut uses dramatic irony to invoke genuine tragedy as well: you can see this at the end of the story, when George and Hazel have watched their son get executed on television and later forget what they have seen.
Finally, Vonnegut makes heavy use of exaggeration in his depiction of characters—particularly in his descriptions of Harrison and the ballerina, who exist as symbols for human excellence and are depicted as blatantly superhuman in their ability. Consider Harrison's size, strength, superhuman intelligence (all at the age of fourteen), as well as the imagery of the dancers leaping up towards the ceiling (a distance of thirty feet). This entire scene has a sense of unreality and distortion to it, reflecting the distorted dystopia in which it is set.