What are three examples of irony in "The Lady, or the Tiger?"
I think that you are probably looking for examples of verbal irony here. This is the type of irony where the author says one thing, but really means the opposite. There is quite a bit of this in the story.
Here are a few examples from the story.
First, the narrator says that the king made
the public arena, in which, by exhibitions of manly and beastly valor, the minds of his subjects were refined and cultured
This is ironic because the minds of his subjects weren't really being cultured by this -- it's barbaric.
Second, we are told that
The arena of the king was built, not to give the people an opportunity of hearing the rhapsodies of dying gladiators...
Dying people don't do rhapsodies. So that's ironic too.
Finally, the narrator says this about the method that the king made up of deciding whether someone was innocent or guilty:
Its perfect fairness is obvious.
This is ironic because the process was ridiculously unfair and arbitrary.