This is a good phrase to practice your sarcasm - a very important aspect of English. It should be used in cases where someone thinks they are telling you some bad news, but it isn't all that bad for you. By saying, "Woe is me!" you will really mean "No big deal" or "Who cares?"
Example: (In this example, you don't like your girlfriend.)
Girlfriend: I'm tired of you always ignoring me and talking to other girls.
You: Well, they're my friends and I can't ignore them.
Girlfriend: Fine! I'm leaving!
You: Oh! Woe is me!
*Door slams.