Mirrors are a big part of any actor's dressing room. Not because they're vain, although they may be, but because it's necessary for an actor to know what she looks like. In this film, the mirrors are all about self-reflection—the psychological kind.
When we're first introduced to Margo Channing, she's looking in a mirror and taking her face off, so to speak. Also her hair. We see Margo stripped down, looking at herself. Margo the woman, not Margo the actress. As we learn, though, Margo has blurred the two. Just as we look at her reflection and wonder who she is, she wonders the same thing.
Mirrors have a similar function in the role of Eve, but with a twist. Critic Kate Bellmore suggests that Eve is a psychopath and that the mirrors help to visually represent her mental illness (source). It's almost as if Eve has to look in the mirror to remind herself who she's supposed to be at any given moment. If the shape-shifter lets her guard down, the whole act is up.
Finally, mirrors play a part in the movie's final scene, which we talk more about in our "What's Up with the Ending?" and the "Mode of Production" sections. The phrase "smoke and mirrors" comes to mind here. In fact, Margo smokes like a chimney, and there are plenty of scenes where she's enveloped in a cigarette haze, too—another way of suggesting her identity crisis.
The mirrors represent one of Joseph Mankiewicz's favorite themes, which the smart folks over at the Harvard Film Archives describe as "the theater as a mirror game of real life in which human identity is revealed to be mercurially unstable, an illusion founded in role-playing and disguise (source).