The primary example of situational irony concerns a world-renowned hunter, Sanger Rainsford, crash landing on Ship-Trap Island, where he is hunted by a maniacal general and becomes the prey. The hunter becoming the hunted is a perfect example of situational irony. Rainsford must rely on his skills as an expert hunter to survive the harrowing experience.

Another example of situational irony concerns General Zaroff's aristocratic, civilized lifestyle compared to his brutal, savage nature. Zaroff is depicted as an educated cosmopolitan who enjoys the finer things in life. He dresses like an aristocrat, reads the works of Marcus Aurelius, and has a refined palate in regards to food and wine. Despite Zaroff's highly cultured taste, he is a maniacal murderer who is obsessed with killing humans on his private island.

Dramatic irony occurs several times in the story and can be found when Rainsford makes dangerous traps that threaten Zaroff's life. The audience knows that Rainsford creates the Malay mancatcher, digs the Burmese tiger pit, and fashions the Ugandan spring trap. However, the unsuspecting Zaroff is unaware of these traps and narrowly survives them. The fact that the audience knows what traps are present while the general doesn't is an example of dramatic irony.

Examples of verbal irony can be found several times in the conversation between the general and Rainsford. General Zaroff utilizes verbal irony when he tells Rainsford that he needed a "new animal" to hunt and refers to it as a "quarry." The verbal irony is that Zaroff is talking about human beings and does not overtly explain that he hunts humans throughout the island. Another example of verbal irony is when Zaroff turns on a light that indicates a false channel and says, "I have electricity. We try to be civilized here" (Connell, 8). This is considered verbal irony because Zaroff is being sarcastic, and his actions are anything but civilized.

There are three types of irony: 

Dramatic: When the audience knows something important that the characters in the story do not.

Situational: When an event occurs in contrast to the audience's expectations.

Verbal: When a character says one thing but means another or means the opposite.

The first post to this question explained an example of situational irony from Richard Connell's "The Most Dangerous Game." There is also an example of dramatic irony in the story, although it is brief. 

When Rainsford first meets General Zaroff, they have a discussion about hunting. Zaroff gives Rainsford some background information about his life as a hunter, including the fact that hunting has become boring to him because he has simply gotten too good at it—it has lost its challenge.

So Zaroff had to find a new kind of game to hunt. As Zaroff tells Rainsford this, the reader begins to understand that he is talking about hunting men before Rainsford realizes it. This builds an expectation in the reader's mind—how will Rainsford react when he finds out? This expectation is the point of dramatic irony. It allows the audience to identify closely with the character as they wait for them to experience a revelation of some sort. 

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.  

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.