Not usually. However, some people do experience leaking while they're asleep. This is because the muscles in the vaginal walls that hold the cup in place relax, so the seal may be broken. Taking care to insert it properly can help.
Since you can wear a menstrual cup for up to 12 hours, most people don't have issues with the cup overflowing overnight. However, you're the best judge of your own flow. You may need to change it more frequently on certain days.
Some people also opt to wear a panty liner or period underwear in addition to the cup.