There are several ironies in Amir’s first story and his telling of it to Hassan. The story is a

dark little tale about a man who found a magic cup and learned that if he wept into the cup, his tears turned into pearls.

First, the story, in some ways, is a metaphor for both Amir's life and Hassan’s. Up until its end, the story represents Hassan’s life. He is the son of a poor man, so he is destined to be a poor man himself according to the restrictive caste society in which they live. Yet he is happy, just as the man in the story is initially happy. Although Hassan aspires to improve himself, as we later see when we learn that he has taught himself to read, he enjoys his life and appreciates the simple things in it.

Another irony is that Hassan cannot understand why the man would need to do something as evil and self-destructive as slaying his own beloved wife. This is in part because of his naiveté and innocence. More importantly, he does not understand how anyone could be so driven to obtain wealth that their need would lead them to commit murder. Hassan is too kind to even contemplate this. For example, when Amir talks him into slinging walnuts at the neighbor’s dog, he is reluctant.

Moreover, Hassan immediately sees the problem with the plot of Amir’s story, asking, "Why did the man kill his wife? Couldn't he have just smelled an onion?" The takeaway to his reaction is twofold. It epitomizes the difference in character between the two boys. Hassan is extremely kind, and Amir is very unthinking and unfeeling at this early stage in his life and in the book. We also see this character difference in the story of the neighbor’s dog noted above.

Finally, the story is also meant as a metaphor for Amir's life. Hassan sees the simple solution immediately, while Amir struggles to find deep meaning in something that need not be complicated. More importantly, Amir metaphorically kills Hassan—or contributes to the killing of his innocence—when he fails to help him in the alley. The event follows him for the rest of his life. He thinks about how he placed the importance of the material item—the kite, represented by the pearls in his story—ahead of his beloved Hassan, represented by the wife in his story. Thereafter, all the success and material wealth in the world cannot make Amir happy until he finally achieves redemption at the end by adopting Sohrab.

What is the significance of irony in Amir's first story in The Kite Runner?

Amir's first story is about a man who discovers that his tears turn into pearls; so he performs acts that cause him to cry, increasing his collection of valuable pearls.  At the story's end, the man sits on a pile of pearls after having killed his wife so that he could cry and produce for pearls. Amir's moral and the story's irony demonstrate that often a person must make great sacrifices at the expense of family and friends in order to get what he wants.

Of course, Amir does exactly what the man in his story did--he sacrifices his friend (and half brother) in order to get his father's approval. Later, Amir also believes that he sacrificed his right to be a father when he betrayed Hassan.  He thinks that his wife's infertility is a result of his heinous act years earlier.

Ironically, Hassan offers a solution to both "stories." For Amir's fictional story, he suggests that the man cut up onions in order to produce tears, and his forgiveness and son Sohrab present Amir with opportunity to redeem himself.