What kind of irony is found in "The Necklace"?

In The Necklace, the irony is called irony of situation.  There is a great difference between what is expected and what occurs.

When Madame Loisel borrows the diamond necklace from Madame Forestier she believes that it is worth a lot of money.  Therefore, when she loses it, she is stricken with the burden of what it will take to replace it, they borrow money to buy a new necklace, effectively giving Madame Forestier a real diamond necklace.  They are then bound to a life of drudgery to pay back the loan.  She does not know, nor do we, the reader, that the necklace is fake. 

Madame Loisel and her husband work very hard to earn money to pay back the loan, believing what they did was noble and honorable.  When they discover that the necklace was fake, they have wasted their time and effort put themselves into a cycle of drudgery for nothing. 

In fact it is ironic that Madame Loisel longs to be rich, a fake existence really, and ends up having her life shaped by a fake necklace. Situational irony.