Give examples of what is meant by "it is impossible to bury the past because the past always claws its way out."

I think the phrase, which occurs right at the beginning of the book, has less to do with time or the past and more to do with being truthful to one’s values and the connections that bind people together. Obviously, the prime example of the past returning is the narrator’s guilt over the fate of Hassan. But Amir is marked by other experiences that endure as well. One example is the conversation Amir overhears as a boy between his father and his friend Rahim Khan, in which his father claims that there is “something missing” in Amir—he never stands up for himself. These words come to define Amir as much as his relationship with Hassan does: although his childhood is defined by his relationship with Hassan, he never considers Hassan a “friend" because the ethnic and religious differences are too great. “In the end, I was a Pashtun and he was a Hazara, I was Sunni and he was Shi'a, and nothing was ever going to change that. Nothing.” (Amir says this in Chapter 4.) There is a sense that Amir’s relationship with Hassan is predetermined by history—that the past will always outweigh whatever personal connection they might have.
In some ways, the book works to undercut the inevitability of history. For one thing, the book shows that the past, or what we think we know of the past, might be fiction. Hassan is not Ali’s son, as Amir has been told his entire life, but actually his father’s. This raises the possibility that other things Amir has been told about himself—like his lack of “backbone,” for instance—might also be false. Nevertheless, these beliefs shape Amir’s life. His decision to return to Afghanistan to claim Hassan’s orphaned son Sohrab is an example of this. In doing so, Amir repudiates his father’s failure to recognize Hassan as his own and his own failure to see Hassan as a friend and equal. At the same time, he is ”standing up” for his values and showing the determination his father always thought he lacked. In this sense, even though he realizes that his father was unfair in his characterization of him, Amir’s actions are still determined by his father’s criticism, although they are transformed by his compassion for Hassan and his love for Sohrab.