The Necklace is a story that can be taken literally or symbolically—it works either way.
Sometimes students have difficulty with symbolism because it involves thinking figuratively. That means you have understand what an author is representing, rather than what the author is actually (literally) saying.
In this story, think about what the necklace causes in the lives of the Loisels. At first, it makes Mathilde feel beautiful. But later, when it is lost, they have to change their lives to pay for Madame Forestier's necklace. Finally, it turns out that the necklace was worth very little. If they had know that, they never would have had to sacrifice themselves to make the money to pay her back.
Now think about what in life affects people this way. In other words, what might make us feel beautiful, or important, but end up consuming our lives in a way that turns out to be wasteful and unnecessary?
The answer to that question should be the answer to your question, what does the necklace symbolize?
I would say that the necklace symbolizes the pursuit of wealth or fame. It makes us feel good for awhile, but it can become burden, eventually coming to dominate our lives in a way that makes us unhappy.
However, it is interesting to note that the writer, Guy de Maupassant, doesn’t leave things quite so simply tied up in a neat little package.
Look at how Mathilde felt in the beginning of the story:
Mathilde suffered ceaselessly, feeling herself born to enjoy all delicacies and all luxuries. She was distressed at the poverty of her dwelling, at the bareness of the walls, at the shabby chairs, the ugliness of the curtains.
It turns out, ironically, that all of the effort that Mathilde had to put into working to repay the cost of the necklace did not make her miserable. On the contrary, the fact that she had to pretty much give up her fantasies of the good life and focus on a life of hard work seems to have made her a little happier, or at least less discontented. This could symbolize the idea that a simple life of honest work is a happier way to live than wishing for things that you cannot have. Late the story, she has become an admirable character:
Thereafter Madame Loisel knew the horrible existence of the needy. She bore her part, however, with sudden heroism. That dreadful debt must be paid. She would pay it.