By definition, "sarcasm" is actually a form of irony that is intended to hurt someone else. A sarcastic remark is "harsh," "bitter," and is generally said with a sneer (Random House Dictionary). We actually never see Juliet use sarcasm in the play, however she does use irony. Ironic words have a double meaning. However, what is most important to remember about irony is that it is expressly used for humor. Ironic remarks are typically witty remarks.
We see Juliet use irony twice in one speech in Act 2, Scene 5. Two ironic statements are found in Juliet's lines,
How art thou out of breath when thou hast breath
To say to me that thou art out of breath?
The excuse that thou dost make in this delay
Is longer than the tale thou dost excuse. (II.v.32-35)
Juliet's first ironic statement points out the ironic truth that if her Nurse were truly "out of breath," she would not be able to say that she is "out of breath." Since Juliet is pointing out a truth that is the opposite of the words "out of breath," she is using irony. The ironic truth Juliet is pointing out is also very amusing and makes the Nurse's character seem even more amusing.
Juliet's second ironic statement is that the thing her Nurse has to relay to her is actually shorter than all the excuses her Nurse is making to delay. Again, this ironic truth is amusing and makes the Nurse's character seem even more ridiculous.