Why is Harrison Bergeron such a threat to society?
A good question. First, let me refer you to the earlier answer by ask966. It makes good points about why Harrison Bergeron threatens his society so intensely.
I would add several other points, though. The first is a variation on the points ask966 made. Harrison is not just above the enforced equality that defines his society. He is in many ways impervious to the restrictions they society places on him. He is weighed down and disrupted, and he still rises above the restrictions, limits, and other handicaps. He literally rises above the handicapped state in the final dance scene.
Second, by demonstrating that he is not equal, he provides a living example of his society's limits. He's like an infection. Everyone who sees him dance will know in a vivid, dramatic, and beautiful fashion that people are not equal.
Because he is heroic, he will inspire others to rebel. He has died, yes, but who knows how many people will be inspired by the memory of his actions.
The memory will also show that the society was unable to anticipate his actions and capabilities. This shows there are limits to the government's power, something that should also nudge people to rebel.
In his short story, “Harrison Bergeron,” by Kurt Vonnegut, the author has developed a society that has tried to make everyone and everything “equal.” At fourteen years of age, Harrison Bergeron is a threat to society because he is clearly above the sameness that the government is trying to enforce through the office of the Handicapper General which means he can question, and doubt, and seek changes that the government doesn’t want. Despite the enormous amount of handicap they have developed for him--huge earphones instead of small ones, glasses with thick distorted lenses that gave Harrison physical ailments like headaches, and huge amounts of metal and hardware hanging everywhere off of him, he is still considered smarter, stronger, more athletic, and etc., and when he escapes from jail the media asserts that he is “under handicapped, and should be regarded as extremely dangerous.”