The main moral or ethical dilemma the story wrestles with is how to mediate or juggle the competing priorities of treating all people with dignity and equality while recognizing that some have extraordinary talent.
The society in which Harrison Bergeron lives has gone overboard in its desire to promote equality. It doesn't want anyone to feel inferior, so it handicaps anyone with a talent in such a way that their talent—or inborn advantage—cannot be used. Physically beautiful people have to wear ugly masks, ballerinas are weighed down with cumbersome weights, and intelligent people like Harrison's father are subjected to loud nosies that prevent them from maintaining a coherent thought.
It is clear that this dystopic society has confused equality of opportunity with equalizing people to the lowest common denominator, no matter what the cost. The beauty, talent, and insight that the most fortunate in a society have and which everyone could theoretically enjoy and benefit from are sacrificed to an ethic of making sure that nobody feels unequal or inferior.
Harrison challenges this ethic by refusing to wear the handicaps and insisting that a ballerina be allowed to dance with him without her handicaps. He is killed for his pains but the story highlights the idea that his society needs to find a better way to mediate the dilemma of meeting both the needs of ordinary people to feel valued and the needs of the talented to excel.