What are some literary devices used in "The Necklace"?

The narrator uses a great deal of imagery in order to show us the life that Madame Loisel imagines for herself. Imagery consists of descriptions of sensory experience, and so it can be visual (sight), auditory (hearing), tactile (touch), gustatory (taste), or olfactory (smell). The narrator says,

She imagined silent antechambers, heavy with Oriental tapestries, lit by torches in lofty bronze sockets, with two tall footmen in knee-breeches sleeping in large arm-chairs, overcome by the heavy warmth of the stove. She imagined vast saloons hung with antique silks, exquisite pieces of furniture supporting priceless ornaments, and small, charming, perfumed rooms, created just for little parties of intimate friends, men who were famous and sought after, whose homage roused every other woman's envious longings.

Thus, we have auditory imagery ("silent antechambers"), visual imagery ("Oriental tapestries lit by torches in lofty bronze sockets"), tactile imagery ("heavy warmth of the stove"), and even olfactory imagery ("perfumed rooms").
The narrator also uses hyperbole, or overstatement, to exaggerate the truth, thereby emphasizing how Madame Loisel feels. He says, "She had no clothes, no jewels, nothing. And these were the only things she loved." Of course, Madame Loisel has clothes and things—just not anything that she thinks of as worthwhile or as beautiful as she would want.
The narrator uses personification when he declares that Madame Loisel "looked at [her husband] out of furious eyes." Eyes cannot be furious, but she is so angry with him over the invitation he brings home that her eyes are characterized in this way. Personification is the attribution of human qualities to something that is not human. Later, her heart is personified as "beat[ing] covetously" when she sees Madame Forestier's beautiful necklace.
The narrator also uses a metaphor when he says that Madame Loisel danced as though she were "drunk with pleasure." A metaphor compares two unalike things; in this case, Madame Loisel's intense pleasure is compared to being drunk.

There is a wonderful example of foreshadowing when Madame and Monsieur Loisel go to the jeweler's shop, hoping to find an exact replacement for the lost necklace. The jeweler recognizes the empty case, but checking his records finds that he hasn't sold such a necklace recently.
This episode ominously foreshadows the climax of the story, when Madame Loisel will realize to her horror that what she thought was the valuable necklace loaned to her by Madame Forestier was actually a fake. This is also a clear example of situational irony, where there is a huge discrepancy between what is expected, and what actually happens instead.
There's important symbolism here, too. The fake necklace symbolizes the phoniness of Madame Loisel, someone who always thought she was better than her modest surroundings, craving wealth and social respectability. And yet because of her greed, her overweening desire to be rich and glamorous, she's ended up as a poverty-stricken housewife, old before her time, much lower down the social ladder than she was before that fateful night when she first put on the (fake) diamond necklace.