The situational irony of the world famous hunter, Rainsford, being castaway on the island owned by Zaroff, a world-class hunter who has become obsessed with hunting humans, is beautifully contrived by the author.
I think the most significant irony is that when Rainsford kills General Zarroff at the end, he is doing so to end the game his own way. He is not playing the game, but in a way he is. The descriptions of the place and General as civilized are also ironic.
One of the chief ironies in Richard Connell's short story "The Most Dangerous Game" is that while General Zaroff displays many of the attributes of a cosmopolitan gentleman, he is in fact a sociopathic murderer. This is a good example of situational irony. In situational irony something in a story occurs that directly contradicts the expectations of the characters or the reader. In this case Zaroff's sophistication initially impresses Rainsford until he learns of the General's "game." Rainsford is shocked to learn that Zaroff is really hunting men. Zaroff lives in a palatial mansion with all the "amenities" of civilization including fine wine and cuisine. At one point Rainsford thinks,
The cocktail was surpassingly good; and, Rainsford noted, the table appointments were of the finest—the linen, the crystal, the silver, the china.
The General is well educated and sophisticated. He tells Rainsford that he reads "all books on hunting" and, after he believes Rainsford has leaped to his death near the end of the story, he is reading the Roman emperor-philosopher Marcus Aurelius:
In the library he read, to soothe himself, from the works of Marcus Aurelius.
This is particularly ironic as Aurelius was a stoic who criticized violence and wrote that ethical behavior was the mark of a great man. Obviously the General thinks himself quite erudite, yet he hunts down men and kills them. For someone to show all the traits of the civilized and still engage in barbaric acts could definitely be considered ironic.
In addition to the ironies mentioned above, the title also presents an example of irony. The double entendre presented in the title is an example of verbal irony, playing on two distinct meanings of the word "game". Game is the animal that is hunted and is also a thing that is played.