I went out of town for 10 days and left my MacBook Pro 13" Unibody (Mid 2010) on. It was closed and unplugged from the wall. When I got back, I didn't mess with it for another 2 weeks or so. When I went to use it, the battery was dead. I plugged in the charger and it lit up green. After a few seconds, the computer came on (gray, faded screen with the progression dots at the bottom). The charger was still green and at the top it said (Not Charging). I unplugged the adapter and the computer turned off. Plugged the adapter back in and turned the computer on. It worked fine. I tested the adapter with my wife's 2009 MB Pro 13" Unibody and it worked fine. After doing a little research, I found out that letting a dead battery just sit like that can really mess it up. Apple's repair would mean sending it to them, waiting for a repair and having it sent back. It would also have been a few hundred dollars more than I paid for the stuff from iFixIt.
The repair was quick and painless and thanks to iFixIt, I had all the tools necessary to make the switch.
Before I swapped batteries, my existing battery started holding a charge, but was only showing a couple of hours, max. I think if this happens with your battery, you can still work with it, but it won't have the same amount of stamina it had before. Leaving your battery dead, but on will definitely mess it up. If you're going to leave your computer for an extended period of time, I'd recommend charging it fully and then just turn it off completely. Also, if you're going to get your computer repaired, I'd suggest looking into the iFixIt stuff first to see if they have what you need and if they have the instructions for doing it. Most repairs only take a few minutes and will save you tons over going the Apple Repair Center route.