The story opens and closes with scenes involving Miss Brill's fur. What is the significance of the fur?

Miss Brill is a very isolated character, detached from the society which she imagines as a "play" she is watching. Even her name, "Brill," suggests a coldness, like a brill fish. Her behavior towards the fur is far warmer and more intimate than her feelings towards any of the human actors in her "play." Indeed, it is almost alive to her, a "dear little thing" which she carefully takes out of its box and brings to life, rubbing at its eyes and smoothing its fur.
The fur necklet, evidently, symbolizes something very important to Miss Brill. First, it a signifier of class to have a fur—the fact that Miss Brill keeps it carefully in a box when not being used, and its faded condition, suggest that she was once wealthier than she is now, and the fur represents a particular level of social standing to which Miss Brill, rightly or wrongly, still clings. Secondly, taking it out clearly represents a special occasion. Miss Brill takes out the fur—her companion and protector of her social standing—just as she takes out herself into wider society.
At the end of the story, then, when Miss Brill puts the fur away, she is symbolically putting herself away too. Returning to her little room, she puts away the image of herself she likes to enjoy when outside, of an elegant lady watching others as one might watch actors in a play. She is putting away the thing that once kept her warm and happy, a symbol of what was once good about her life. The worst part for Miss Brill is that she has been exposed—the fur does not make her look as she had hoped in the eyes of others. A young girl, giggling, has described her fur as looking like a "fried whiting." When Miss Brill imagines she hears the fur crying, then, as she puts it into its box, we can interpret this as her own muffled cry for help. The young girl has exposed the fur for what it really is: a sad, bedraggled, old thing past its prime. Miss Brill recognizes, on some level, that she herself feels this way: she and the fur, trapped in the confines of their respective boxes, are one and the same.