In the first paragraph, the narrator notes that Mathilde is a charming girl and "by a mistake of destiny" was born to a family of clerks. The suggestion is that usually such a charming girl is born to more wealthy families. So, the actual result (that she was born into a poor family) is ironic considering that such a girl is usually expected to come from wealthy families. (This would be situational or cosmic irony.)
Verbal irony is usually defined as speech or language that means something other than which it seems to express. This is often confused and/or compared with sarcasm. For example, saying someone is "as strong as a mouse" is verbal irony because although the words used seem to express strength, the meaning is actually the opposite. There seem to be no clear examples of verbal irony in "The Necklace." The characters are very direct in what they are saying.
Dramatic irony occurs when the spectator or reader is given some information about which one or more characters is/are unaware. Mathilde feels like she should have born into a richer life and she continues to crave that life despite being poor. However, this is not really dramatic irony because the reader and the character are aware of the same things. If anything, this is an example of situational irony or maybe even cosmic irony if the reader (or Mathilde) considers that God or some spiritual power has deemed her to be born into poor circumstances yet with a craving for a richer life, thus the gods are toying with her.
Even the ironic twist at the end of the story is not verbal nor dramatic irony. The statement is direct, not misleading. And the reader discovers the ironic twist when Mathilde does, not before she does, so it is not dramatic irony. ("Oh, my poor Mathilde! Why, my necklace was paste. It was worth at most five hundred francs!") Again, the ironic twist at the end would be an example of situational irony. Mathilde had originally been enamored with a cheap necklace, not noticing that the diamonds were not real. There are no overt clues to the reader or Mathilde that the necklace she lost was cheap. So, the outcome (discovering that the necklace was cheap after all that effort to replace it) was unexpected and therefore an example of situational irony. (The only possible clue, and therefore evidence of some vague dramatic irony, is that Mme. Forestier was so easily willing to lend Mathilde the necklace in the first place.)