Who is actually running the government in the story "Harrison Bergeron"?
The text of the story never actually addresses who is running the government.
The action takes place in the United States, and the Constitution is still apparently the structural basis for the government. This would lead to the assumption that the citizens of the United States still voted to elect governmental officials.
However, the office of the Handicapper General apparently has acquired enormous influence, as the agents of that body are in charge of insuring that "Nobody was smarter than anybody else. Nobody was better looking than anybody else. Nobody was stronger or quicker than anybody else."
The equalizing impact of the handicapping devices distributed and enforced by the Handicapper General and her officials would have rendered all political candidates as being "equal every which way." Since all candidates were handicapped to mediocrity by devices determined in response to their particular areas of exceptionality, electing any one candidate over another probably would not have greatly impacted the actual function or power of the government.
Looked at from that standpoint, it could be argued that Handicapper General Diana Moon Glampers was the head of the government.