What seems especially ironic in Death of a Salesman is that Willy Loman is a failure but he keeps advising his sons about how to be a success. He was advising them how to be a success when they were kids in school, and he is still advising them how to be a success when they are in their thirties. In the meantime, Willy has becoming less and less successful because he is getting old and worn out. Willy doesn't know how to succeed. In his imagination he keeps asking his brother Ben the secret of success, which shows that Willy doesn't know the secret, if there really is a secret. Willy cannot tell his sons how to succeed and he cannot serve as a role model of success because he is far from being a success himself and doesn't even know how he could have succeeded. It is also ironic that Ben can't tell help him with advice. Ben keeps saying that he walked into the jungle and when he came out, by God, he was rich.
There is also irony in how the Lomans regard Bernard as a young man. He is mocked for trying to get Biff to study so he will graduate-
WILLY: Don't be a pest, Bernard! [To his boys] What an anaemic!
Willy goes on to qualify his criticism of Bernard by explaining that his lack of personal charisma will hold him back -
WILLY:...Bernard can get the best marks in school, y'understand, but when he gets out in to the real world, y'understand, you are going to be five times ahead of him...Because the man who makes an appearance in the business world, the man who creates personal interest, is the man who gets ahead.
In Act 2 we see the grown up and successful Bernard visiting his father before he returns to his work. It is Charley's turn to boast, and yet he does so gently-
CHARLEY [An arm on Bernard's shoulder] : How do you like this kid? Gonna argue a case in front of the Supreme Court.
We see the irony in Willy's inability to read or direct the future of his own sons, nor to predict the potential of his nephew.
One of the ironies in the play involves the notion of the "American Dream." Willie longs to live this dream; however, Willie is living an illusion. The skills he learned in the 1920's as an up-and-coming salesman are now obsolete. Now, the job market requires "specialized skills and knowledge, and because of this, Willie is doomed to failure" (Enotes). The irony is that Willie keeps believing that he, in fact, CAN achieve this "American Dream," but the reader knows differently. He cannot ever be the man he longs to be; therefore, he is a complete and utter failure. Another instance of irony is in Biff and Happy, Willie's sons. They both, in their early years, appeared to be able to one day achieve success. Biff was a star athlete and Happy was more grounded than his father was about his career choices, etc. However, Biff never takes advantage of his opportunities and has the same grandiose visions of success without anything to "back" them up; he also doesn't possess the skills to have a rewarding job or career. Happy, on the other hand, seemed to be more grounded than his father and Biff; he seemed to possess more reason and logic when it came to his career and life; however, by the end of the play, he has also fallen victim to the grandiose visions of the "American Dream" and vows to become what his father dreamed of. There are more ironies, as well, that others may point out.
When they say "we're free" it is a very important part of irony because they are finally financially free, however, Linda is also free from being dominated by Willy. Another one is when Biff goes to see Charley and sees his son Bernard. Even though Willy thought Bernard was going to be a nerd, Bernard is happy and financially secure with kids, while his owns son do not have it together. This showcases how things will turn always turn out as you expect since Willy believed Bernard would always be a nerd, but in fact Bernard has his life set more than his own sons. On top of that, Charley knows what his son is doing and how he is doing it while Willy is unaware of his sons plans. This shows irony at Willy's thoughts versus the reality.
The irony is that he committed suicide on the day their home which he spent a lifetime paying for at long last become their home.
Just some pretty basic ones-
Willy's last name "Loman" pronounced Low-Man suggest his low place in society though he insists otherwise.
Linda's last line "we're free"- ironic because they are finally able to pay off all debt and be free of any burden only due to Willy's killing himself for the insurance money.
"We're free." is ironic because apart from being financially free, Linda is free from being dominated by Willy, too.