Other contributors have already spoken about the major ironies of "The False Gems": that Monsieur Lantin had assumed his wife's fidelity when, in fact, she had been in all likelihood unfaithful; that the jewelry he believed had been false had been real all along; and finally, when he married a second, virtuous wife, only to be made miserable in the process. To this, I would add a few smaller ironies, interwoven among those major ones.

For example, take the following passage, as written by Maupassant:

Sometimes, of an evening . . . she would place on the tea table the morocco leather box containing the "trash," as Monsieur Lantin called it.

The trash, of course, refers to the presumed false jewelry. While this sentence ultimately reads back into the larger irony relating to the false jewelry actually being real, there's something deeply ironic in the thought that he would label these gemstones specifically as "trash," when they are, in fact, worth a fortune. In fact, this "trash" will later make him very wealthy once he learns the truth about them.

In addition, when he resolves to sell the jewelry, he does so because "the very sight of them spoiled, somewhat, the memory of his lost darling." In the process, however, he uncovers a far greater dishonesty at play, which overturns all his assumptions concerning his wife.

The irony in "The False Gems" by Guy de Maupassant is that what Monsieur Lantin expects to happen never does. Instead, his expectations are subverted again and again. Irony makes up the bulk of the narrative, as Lantin moves from one false assumption about his world to another. 

Monsieur Lantin doesn't like the jewelry his wife wears and assumes it's fake. He patronizes her, suggesting that she has bohemian tastes and should go out without jewelry. He refers to her jewelry as trash; it isn't until she's dead that he learns the jewels were real. 

When he realizes the true value of the jewels, he assumes his wife was unfaithful and received them as gifts. He goes home and cries himself to sleep. He was unaware that she wasn't what he believed her to be. 

Lantin is able to live a more privileged life with his money. He quits his job and begins to enjoy the theater for the first time. When he marries again, he chooses someone who is a "very virtuous woman." De Maupassant says only that she had a very bad temper and caused him much sorrow.

All of these are examples of situational irony. Situation irony is when an outcome is not what a character—or the reader—expects. Lantin thought his wife was a virtuous and faithful woman who loved the theater and fake gems; instead, she was likely unfaithful and wore real, very expensive, jewelry. Lantin expects that the fake jewelry will fetch a low price; instead, it fetches a very large sum. He expects that a second, virtuous wife will make him happy; instead, she makes him miserable. 

The story that you refer to is "The False Gems," in which the main character M. Lantin is married to a woman that he truly loves, he adores her, as the years pass in their marriage he loves her more and more.  There are only two things that he does not like about her.  She loves to attend the theater and has a passion for fake jewels.

The irony of the story, situational irony, where an outcome is completely different that what is expected is evidenced in the "false gems."  Also, verbal irony, M. Lantin's wife knows that the gems are real, but she pretends that they are fake.

After his wife dies, his life is miserable.  He can't stand the sight of the gems that his wife brought home constantly.  By now he is poor, destitute.  So he decides to sell the gems.

The irony is that when he tries to sell them, he learns that the gems, the jewelry is in fact real and very valuable.  M. Lantin sells the gems for a great sum, now he is wealthy.

His wife must have had an admirer, a rich one, who gave her the gems as gifts.  So in fact, his loving, attentive wife, was probably having an affair.  Yet he never knew it, he was shocked.

M. Lantin decides to remarry, his second wife is very virtuous, she will be a faithful wife. 

The irony is twofold, M. Lantin never knew that the gems were real and that his first wife was technically, false.

The second wife, who should make him happy with her fidelity, makes him miserable.