What role does the diamond necklace play in the story?
I am assuming that you are referring to "The Necklace" by Guy de Maupassant. The necklace in the story is actually two necklaces, the first being the fake diamond necklace Madame Loisel borrows and loses and the second being the real one the Loisels purchases for its replacement.
Both necklaces operate as symbols in the story. The necklace that Madame Loisel borrows is a symbol of her yearnings to be wealthier and more important than she is. Because it is a fake necklace, one can reasonably interpret the necklace to show that these are yearnings not worth having, based on sparkle and no substance or value. The fake necklace is a "lie," just as Madame Loisel's attempting to be something she is not is a lie and as her refusing to tell her friend the necklace is lost is a lie. The real diamond necklace, on the other hand, the one she replaces, is acquired through hard work and deprivation, of far greater value to anyone in the long run than lies and misreprentations.
To some degree, for me, these necklaces evoke the glass slipper in Cinderella. Madame Loisel dons the fake necklace to go to the ball, but since, of course, this is not a fairy tale, but a story with a message about dishonesty and vanity, when she loses the fake necklace, there is no prince to retrieve it and find her, and she and her husband must rescue themselves from vanity and lies through years of hard work.