In the story, Harrison is handicapped to make him as "normal" as most other people. This might have been done in the spirit of equality, but it homogenizes an entire community. It is an infringement on human freedom and since the more intelligent individuals are handicapped, this hinders the potential for technological and even social innovations and progress. So, one moral or suggestion that's presented here is that we must question the ways in which we strive for equality. One suggestion is that suppression is not the way to achieve equality. In other words, equality is meaningless if everyone is basically identical. Equality only means something if we are diverse and then treat each other as equals.
The subtext of "Harrison Bergeron" is the extent of this suppression. George and Hazel, Harrison's parents, are pacified by their television. Note that Harrison uses the television to announce his expected Emperor status. Harrison recognizes that television is his best way to communicate. Although his freedom is a liberation, he turns it into an opportunity to become a dictator. The point is that television is used by Diana Moon Glampers and Harrison to advertise their power and ideologies.
Suppression of mental and physical abilities can come in the form of physical/mental restraints. But television and other ubiquitous media sources also wield a large sphere of influence over the public. So, one moral is that the pursuit of equality has meaning only if it is built upon diversity and does not involve suppression. The second moral is more of a comment on the dangerous power that social institutions (television, media) have in conditioning the populace.
This is especially relevant in an age when we are bombarded by media via television, Internet, and mobile phones. This includes the often discussed problem of how we've become desensitized to violence.
George and Hazel were watching television. There were tears on Hazel's cheeks, but she'd forgotten for the moment what they were about.
Such a profound suppression of people's individuality and humanity could be perpetrated by any number of institutions from government to large corporations to social media. The irony is that these institutions can be used for the opposite purpose: to foster freedom and creativity.
The moral of Kurt Vonnegut's "Harrison Bergeron" is that forced equality--"It was the year of 2018 and everyone was finally equal"--is not truly equality; it is forced mediocrity. For, in order to place everyone upon an level plane, the brillant, the creative, the talented must be suppressed. Indeed, those superior to others must be brought lower since it is impossible to raise those without capabilities to a higher level.
Hazel, who represents the "equality" level wears no handicaps or other devices, for she cannot be raised from her level of mediocrity. Rather, it is the super sensitive, highly intelligent, athletic, and handsome Harrison, his brillant father George, and the extremely graceful and beautiful ballerinas who are loaded down with handicaps. Thus, the suppression of their superior capabilities dumbs down all of society to the level of the banal.
In addition to this suppression of those who are superior to others and their reduction to mediocrity, there can also be other detrimental effects, such as rebellion as exemplfied in Harrison's character, and the loss of initiative as evidenced in George who becomes afraid to exercise his intelligence, thus accepting his mediocrity,
"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 everbody else."
Clearly, George has lost his initiative and in 2018 beauty, glace, and wisdom have died, shot by the Handicapper General who insists upon mediocrity and the suppression of the individual to the point of murdering people. Thus, the "equality" of Bergeron's world is no equality at all. Mediocre at best, it verges on the sadistic.