Why is the government looking for Harrison?

There are four reasons the government is searching for Harrison Bergeron. The most immediate reason the government is looking for Harrison is that he broke out of jail. The second reason is that they suspected him of planning to overthrow the government. 
The next reasons the government is looking for him are specific to the future society Vonnegut evokes. The third reason is that Harrison is "under-handicapped." Under this future American government, people whose abilities or gifts raise them above average have to be handicapped. He isn't handicapped enough. And that leads to the final reason they want Harrison: he is an active, living, and vividly dramatic challenge to the government's conceptual and ethical foundation. Just by living and being strong, smart, and handsome, Harrison threatens them.

Harrison Bergeron, the fourteen-year-old son of George and Hazel Bergeron, had been arrested in April 2081 and put in jail. He was suspected of trying to overthrow the government, although how he would have done so or what he actually tried to do is not overtly described in the story. As George and Hazel are watching TV, they hear an emergency announcement stating that Harrison has escaped from jail, and his picture is flashed on the screen. Soon Harrison himself appears live on TV. He has torn a door off its hinges and proclaims himself Emperor. He then begins to remove all his handicaps--scrap-iron, headphones, wavy-lensed spectacles, and red rubber ball nose. He then chooses a ballerina for his Empress and commands the orchestra to play beautiful dance music. As he dances with his Empress, allowing people to "watch what [he] can become," Diana Moon Glampers, the Handicapper General, appears in the studio and executes him and the ballerina on live TV.
The government is looking for Harrison because he has escaped from jail and threatens to overthrow the government by showing everyone what human beings are capable of when they allow themselves to embrace their strengths rather than squelch them for the sake of equality.