Dramatic irony occurs when the audience (or reader) knows something that one or more of the characters do not. In other words, the audience is one or more steps ahead of the characters in the story. There really are no clues or hints that give the reader this kind of advantage over Madame Loisel, her husband, or any of the other supporting characters. The reader and Madame Loisel learn the necklace is made of fake diamonds at the same time.
There is, however, evidence of situational irony. This is when the outcome is different or opposite of that which was intended. Madame Loisel expected the necklace to bring her happiness. It did for a short time, but then losing the necklace made the rest of her life quite difficult. She expected that the necklace was made of forty thousand francs worth of diamonds. She later learned the diamonds were fake. Earlier in the story, Monsieur Loisel expects his wife will be thrilled about the invitation to the Ministry. She is initially too ashamed to consider going because she feels she requires expensive clothes and jewelry to even fit in with such a high strata of society.