You definitely need to drain off the extra if it's overfilled. Why do you think it's overfilled?
What procedure was used to fill the gearbox? Are you sure the correct fluid was used? We'd need to know that to know if you need to drain completely and refill.
On a manual, it would probably get a little frothy, maybe get some gears hot (binding shifts), spill out the breather hole(s).
On an automatic you're looking at an electronic tranny calling in sick or going to limp home mode, but otherwise you're looking at hard shifting, overheating, blown seals- basically, dollar signs.