The main reason Harrison Bergeron is considered a threat in the eponymously titled short story is that he is able to rip off of his handicaps and make himself the self-proclaimed "Emperor," thereby removing control of the authoritarian regime that needs all of its citizens to weaken themselves and obey government commands.
The people in 2081 America willingly give in to the government, which asks those who are "well-above average" to reduce themselves intellectually and physically by installing both mental and physical handicaps on them. Early in the story, George Bergeron scolds his wife after she suggested he "take out a few of them lead balls" that make up his physical handicap. He tells her, "If I tried to get away with it ... then other people'd get away with it-and pretty soon we'd be right back to the dark ages again, with everybody competing against everybody else."
But Harrison does more than "take out a few of them lead balls" to his handicaps. He "tore the straps of his handicap harness like wet tissue paper" and "snapped" his mental handicaps that fit over his ears "like celery." Both of these handicaps were "guaranteed to support five thousand pounds."
When Harrison declares himself emperor and names his empress, he begins to order those around him to defy the government. This is when Diana Moon Glampers, handicapper general, enters the room and shoots Harrison dead.
In a society that needs order, ignorance and control, a person who defies all three becomes the greatest threat.
Harrison is a threat because he is attempting to reinstate his individuality which the state has plotted to remove from all citizens by enforcing handicaps to make everyone "equal."