How does the dollhouse symbolize class discrimination in "The Doll's House"?

The dollhouse is a symbol of privilege among the upper class. The Burnell sisters receive the dollhouse as a gift and are permitted by their parents to show it off at school. This only heightens their popularity, since they come from a rich background. However, as the schoolgirls gather around the dollhouse, there are two poor girls that are forbidden from viewing the dollhouse. These are the Kelvey sisters, daughters of a washerwoman and a rumored criminal. Their poor background prevents them from experiencing the same level of privilege as the other girls.

This sets up the main conflict of the story. Class discrimination is presented in something as trivial as a dollhouse. Despite the warnings from her parents, Kezia Burnell allows the Kelvey sisters to view the dollhouse anyway. This act of rebellion can be seen as a condemnation of class discrimination because it goes against what is normally accepted in the upper class. When Kezia shows the Kelvey sisters the dollhouse, she is immediately berated by her aunt. Once the Kelvey sisters escape, Else mentions that she has seen the lamp in the dollhouse. She had a chance to experience something that was once prohibited from her.

Can the doll's house and the dolls in "The Doll's House" be seen as symbols of the upper and lower classes respectively?

Your argument on how the Burell's doll house represents the upper class society is certainly accurate. There are several examples from the story that we can extrapolate to demonstrate exactly how this dollhouse symbolizes a separate part of society.

First, notice the opulent manner in which Mansfield describes the dollhouse. Not a detail is spared in the decor, the objects within it, and the overall look of wealth that the house inspires.

There were pictures on the walls, painted on the paper, with gold frames complete. Red carpet covered all the floors except the kitchen; red plush chairs in the drawing-room, green in the dining-room; tables, beds with real bedclothes, a cradle, a stove, a dresser with tiny plates and one big jug.

Such a dollhouse cannot belong to someone any less able to appreciate its delicate features. Moreover, the gift of the dollhouse itself is described as "most sweet and generous", as if the expense of this gift is taken for granted. It is clear that the appearance of the house, as well as the fact that only a selected few of the Burnell girls' friends could go to see and appreciate it, makes it a very exclusive thing that very well mirrors the upper echelons of high class society.

Another thing to consider is the way in which the Burnell girls are treated as school celebrities because of the dollhouse. The girls at the playground put their hands around them, talk to them, and ask for turns to see the dollhouse. This also shows the dollhouse as a symbol of class separation.

In an ironic twist, Mansfield offers this added detail to the description of the dollhouse right before describing the two poor, disliked little Kelvey girls:

The father and mother dolls, who sprawled very stiff as though they had fainted in the drawing-room, and their two little children asleep upstairs, were really too big for the doll's house. They didn't look as though they belonged.

The two Kelvey girls, like the two big dolls referred to above, are equally unfit to penetrate that superior world created by social class. They had no known father, their mother is a washer woman, and their clothes are obvious hand-me downs that make the girls look unkempt, and basically neglected. They are humilliated by the other girls in the playground, and left to their own devices to make it daily as best as they can. As a rule, the Burnells are not allowed to invite the Kelvey girls to their home, not even to see the dollhouse. Even when they are able to sneak-in and take a look they are treated as if they were contagious

"You know as well as I do, you're not allowed to talk to them. Run away, children, run away at once. And don't come back again," said Aunt Beryl. And she stepped into the yard and shooed them out as if they were chickens.

Therefore, with these examples we see how the dollhouse definitely symbolizes class separation; one which favors some extremely well while the rest are unable to get a go at it. Ultimately, this is what prompts Else to feel so much joy for having had a glimpse of the famous lamp of the dollhouse. It was her first and last chance to see what "the others" see.