The most important example of irony is that Mathilde had spent years of hard work to make up for the money it cost her to replace her debt (for buying a diamond necklace) when in fact the necklace she borrowed and lost was a fake necklace.
In the beginning of the story, Mathilde refused to go to a party because she didn't have anything to wear, let alone any jewelry to accent it with. After she borrows the necklace, she loses it somewhere on her way home from the party. Had she told the truth, Mathilde would have found out that the necklace was not genuine. Instead, she went out and bought a real diamond necklace to replace the one she borrowed.
Another example of irony is that Madame Forestier didn't even look at the necklace that was returned after Mathilde borrowed it. Mathilde was worried that if she looked at it, she would see that it wasn't the exact same necklace. The irony here is that if she DID look at it, she would have seen that it was a genuine necklace, not the fake one she owned. All would have been resolved at this point instead of years later after the "social climbing" Mathilde dropped her status by having to do hard labor.