What are the burdens that Holden carries?

You are correct. Holden feels burdened over his brother's death. Perhaps the greatest theme of the novel involves the relationship between the pain of actual experience and feeling one's feelings, on the one hand, and on the other hand the equally devastating numbness that comes with shutting down one's emotions in order to avoid suffering. After the death of Allie, Holden essentially shuts down, forcing himself to lose all attachments to people so as never to be hurt again. He repeatedly mentions how important it is not to get attached to anyone, since this will lead to missing them once they are gone. By the end of the novel, he has spiraled so far down with this theory that he has become afraid to even speak to anyone. Phoebe is perhaps the only reminder that Holden still has the capacity to love.