An additional irony (additional to the one pointed out by mwestwood above) comes in the observations made by a tourist couple in the closing passage of the novel.
Sitting at the Terrace, the tourist spot the bones of the great fish that the old man, Santiago, caught and brought back (dead and decimated) to shore. They ask what it is and, failing to listen to the answer the waiter was prepared to give, conclude that the great dead fish is a shark.
"I didn't know sharks had such handsome, beautifully formed tails."
This line presents a number of ironies. First and most superficially, the tail is the most identifiable detail of the fish that Santiago has caught. The fish is a marlin, best known by its tail. Secondly, the marlin caught by Santiago was eaten by sharks and so is made unrecognizable by the very type of fish that the tourists mistakenly believe the marlin to be.
The more substantial irony of this moment comes in the idea that Santiago's accomplishment of catching such a gorgeous and grand fish is not clearly acknowledged by these people who do, nonetheless, manage to see that the fist is in some ways impressive.
Like Santiago himself, the fish is no longer identified or known for its greatest attributes though these attributes are still present. Santiago's strength of will prevails in his struggle with the great fish. For the fish, even the skeleton of the tail is enough to impress the tourists.
A connection between Santiago and his great catch is thus ironically solidified by people who never even see Santiago. His prowess as a fisherman is proven by the observation made by the tourists as they see only an echo of the skills and strengths and significance of Santiago's epic fishing day.