The American dream has a generally negative effect on Willy, in his psychology and in his life. Willy becomes bent on achieving a dream of success that is beyond him and this leads him to ignore his actual qualities and his moderate successes.
Though Willy's brother Ben achieves the success that Willy grasps after (in conversation, mostly), Willy is unable to act as Ben does, boldly, confidently, etc.
For Willy, the American dream consists of two things - being "well liked" and acquiring real wealth. Though Willy and Linda manage to pay off their house, an accomplishment, they do not get rich from Willy's time on the road as a salesman.
As the play progresses, we see that Willy's chance to achieve the success he desires is impossible. He is not a good enough salesman to get rich "selling" and never was. His dream is a false dream. His dream is fine for someone else, but does not fit his stature intellectually or emotionally.
His dedication to this dream turns into delusion as evidenced by his hallucinatory conversations with Ben. Finally, this dedication turns into destruction as Willy grasps at an ultimate solution to the problem of his failure to succeed.