No, Lady Capulet does not die, at least, not during the action of the play. But after discovering what appears to be the dead body of her daughter, she's understandably grief-stricken, so much so that it wouldn't surprise us if she took her own life in response.

Although Juliet isn't actually dead at this stage in the play, Lady Capulet thinks that she is, and so her heartbroken response indicates absolute sincerity in the expression of her emotions. And when Juliet is found dead later on—this time for real—if anything Lady Capulet's reaction is even more intense. She may not die herself in the play, but something inside her undoubtedly does after she comprehends the full tragic horror of her beloved daughter's death.

Juliet's mother does not die in Romeo and Juliet. She is at Juliet's tomb in the final scene, when all is revealed, and the Capulets and Lord Montague discover that their children have died because of their old family feud. Lady Montague, however, does die. When he arrives at the Capulet family crypt, Montague tells the Prince:

Alas, my liege, my wife is dead to-night!
Grief of my son's exile hath stopp'd her breath.

Lady Capulet's final lines, however, suggests she feels like she will die of grief as well. She says the sight of her daughter's body next to Romeo is "as a bell that warns my old age to a sepulchre." She is obviously devastated by her daughter's death (which she has to endure a second time, as they thought her already dead), but she is still alive at the end of the play.