Chekhov certainly relies on dramatic irony in "The Bet" because the story is told mostly in flashback. As it opens, the banker remembers back to the discussion he participated in with the young lawyer 15 years earlier when he made the bet with the lawyer. The banker knows and the reader eventually discovers that the banker does not have the funds to cover his bet should he lose. Chekhov, however, uses dramatic irony by providing that information to his readers but not to the lawyer.
Situational irony plays a more significant role in the story. For example, the lawyer quickly argues that life in prison would be much better than a criminal receiving the death penalty, but in the end after completing his 15-year experiment with "imprisonment," he comes to hate life and solitude and has nothing to live for. If the lawyer were able to have the same conversation with the banker after his imprisonment, he would have argued that the death penalty would be a more humane punishment.
Another example of situational irony occurs when the lawyer leaves just 15 minutes shy of his 15-year "sentence." One would think that he would just complete his bet and take the money he earned, but he intentionally escapes before the bet is fulfilled. Not only is that aspect ironic, but Chekhov also employs irony when the banker just happens to show up when the lawyer is sleeping and is able to read the lawyer's letter. The letter which describes the lawyer's disdain for his life and essentially his wish to die is actually what prevents the banker from making the lawyer's wish come true (he was plotting to kill the lawyer so that he would not have to pay his debt to him). So, in the end, the lawyer who wants to die unknowingly convinces the man who was planning to kill him not to take his life!