How does Vonnegut's story, "Harrison Bergeron," compare to Plato's Allegory of the Cave?

The short answer is that the characters in Vonnegut's story find themselves in a situation very similar to the one described in Plato's scenario. They are prisoners in a carefully restricted, artificial reality. Ordinarily, their freedom is restricted so much that they can't even perceive clues that another, greater reality exists. When they are confronted with clear evidence from a prisoner who has escaped and discovered the truth, the prisoners remain unconvinced. Pain and the misleading nature of their perceptions prevent the prisoners from attempting escape.
To appreciate these correspondences, let's first take a look at key elements in the Allegory of the Cave. Then we'll consider the specific ways in which "Harrison Bergeron" matches up.
In Plato's allegory, people are held prisoner and allowed to look only at the wall opposite them. They have lived there since they were children, chained and shackled, and cannot move around or look behind them. The only action they see is the movement of shadows across the wall. These are produced by people and objects moving behind them, illuminated by a fire. The prisoners have no way of knowing what causes the shadows they see, so they take them at face value. They don't suspect they are the signs of something else — the existence of a larger world.
What, asks Plato, might happen if one of the prisoners were allowed to turn around and look at the fire? Because his eyes were accustomed to darkness, the light would hurt his eyes and he would look away. He would likely choose to resume his former ways (staring at the wall with the shadows), and he would resist the idea that the fire represented true reality. It is visually easier to look at the shadows, and they are familiar, so he accepts them more readily as reality.
Similarly, if a prisoner left the cave, he'd be overwhelmed by the painful, bright light. If he remained outdoors, his eyes would eventually adjust, and he would be able to see clearly what the real world looks like. Over time, he would figure out how shadows work, and realize how the subjective experiences of the prisoners have misled them about reality — a wider world exists beyond the shadows.
If the escaped prisoner returned to the cave to liberate the others, though, he'd immediately be blinded in the darkness. Unable to see anything (because his eyes haven't yet readjusted), he would appear to the other prisoners to have been harmed by his experiences. Therefore, they wouldn't be receptive to his message about the advantages of quitting the cave. They would fear leaving, and even fight to stay.
That's the Allegory of the Cave. How is "Harrison Bergeron" similar to it?
1. The people in the story inhabit a world where the government carefully controls what they can see, and they are compelled to wear devices and shackles that prevent them from perceiving the larger reality.
In Vonnegut's story, people are prisoners of a police state. The United States Handicapper General monitors each individual to make sure that he or she has no competitive advantage over anyone else. If a person has more than his or her share of a valued trait, this advantage is neutralized by devices, weights, and shackles.
For instance, George, whose intelligence is "way above normal," has a "handicap radio in his ear" that blasts painful, disoriented noises every 20 seconds. This prevents him from maintaining a coherent train of thought, and appears to interfere with his memory processes. Two of the eight ballerinas wear the same device, and Harrison wore earphones that function like George's radio. Apparently, then, a sizable fraction of the population has their thoughts controlled in this way. In addition, people wear onerous weights — like the "forty-seven pounds of birdshot in a canvas bag" hung around George's neck — that interfere with their ability to work, move with grace, investigate the world, or otherwise resist the government's control. Kept under control by these means, people allow themselves to be convinced that removing handicaps would destroy society.
2. Harrison Bergeron is analogous to the prisoner who has escaped Plato's cave, learned the truth, and returned to free the other prisoners.
When Harrison escapes from jail, he doesn't attempt to leave his society (the allegory of the cave). Instead, he goes to the heart of this society's mass communication system — the television studio — and removes all his physical handicaps on camera. Much of his motivation might be selfish. He announces that he is "a greater ruler than any man who ever lived" and makes it clear he intends to use his advantages to be an autocrat:

"I am the Emperor!" cried Harrison. "Do you hear? I am the Emperor! Everybody must do what I say at once!" He stamped his foot and the studio shook.

Harrison wants to enjoy the beauty and talents that the government has hidden with shackles, masks, and weights. He strips the ballerina and musicians of their handicaps — freeing these prisoners from the conditions imposed by the cave — and he does so in front of all the television viewers. So his actions present people with the reality that had been hidden from them.
3. Just as Plato's prisoners reject the truth after encountering the returning prisoner, so do the people in "Harrison Bergeron."
Diana Moon Glampers kills Harrison and the ballerina, and orders the musicians to put their handicaps back on. The television burns out, and George is hit with an especially painful, forceful blast ("a doozy") that wipes out his memory. Hazel, who has been crying, can't remember what caused it, except that it was "something real sad on television," and both she and George agree that she should "forget sad things." Like the prisoners in the cave, they associate hints of escape with something painful, so they are content to remain in their situation, ignorant of the truth.
http://classics.mit.edu/Plato/republic.8.vii.html