What is the irony in "The Gift of the Magi"?

Irony is a literary device that occurs when the author uses elements of the story, like plot or character development, setting, or action to show contrast or the opposite of what the characters or the author may be feeling. 

Here's an example--  In "Rhyme of the Ancient Mariner," there is a line that says "Water, water every where, not any drop to drink."  This is an example of irony, because the ship is in the middle of the ocean.  Yes, the crew is surrounded by water, but they can not drink any of it.  Ah, the irony!