What is the dramatic irony in "The Story of an Hour"?

Dramatic irony is evoked when the reader knows more about something than the characters in the story do. There are two key elements of dramatic irony in this story. Because a significant portion of this story focuses on Louise Mallard when she is alone in her room, the reader is privy to her innermost thoughts during this time, while Josephine and the others do not know what is going on in the room. They have not witnessed the young widow—as she believes herself to be—beginning to feel that she is "free" as a result of her husband's death. Therefore, when Josephine begs Louise to open the door, fearing that she will "make yourself ill," there is an underlying dramatic irony. Josephine is afraid that her sister is in hysterics of grief, whereas the reader knows the opposite to be true.
The second key element of irony comes at the very end of the story. It is broadly ironic that, instead of being freed by her husband's death, Louise is actually killed by shock at his reappearance. Specifically, there is dramatic irony in the fact that the doctor declares her dead of "the joy that kills." In actuality, we know that Louise had just come around to feeling happiness at being left without a husband. As such, we can be fairly sure that it was not joy she felt at the reappearance of her husband. Instead, his reappearance has killed her freedom and her prospect of joy itself, causing her to die of disappointment.

Dramatic irony refers to when the reader or audience knows something that one or more characters do not.  In this particular story, we know that Louise Mallard is not in her bedroom, grieving and inconsolable, though her sister is unaware and is truly concerned for her health and well-being.  

Josephine was kneeling before the closed door with her lips to the keyhole, imploring for admission. "Louise, open the door! I beg; open the door—you will make yourself ill.  What are you doing, Louise?  For heaven's sake open the door."

Josephine assumes, quite naturally, that Louise is grief-stricken about the loss of her husband because this is the normal reaction to have when one's husband dies (and we know that Brently Mallard was a good husband because Louise thinks about how he "had never looked save with love upon her").  In reality, however, Louise is actually experiencing a "monstrous joy" at the prospect of "no one to live for her during [the] coming years."  She is thrilled that she will now be free: she whispers, again and again, "Free!  Body and soul free!"  Thus, we know more than Josephine and Richards, Brently's friend who came to deliver the news.
 
In the end, when Brently returns and Louise dies at the sight of him, we also realize that the cause of death decided on by her doctors, "joy that kills," is incorrect.  Louise does not die of joy when her husband returns; she likely dies of disappointment: all that freedom she so looked forward to enjoying vanished the moment she realized her husband was still alive.