Close All Open Apps... Perform a Reset... Try again...
Reset ( No Data will be Lost )
Press and hold the Sleep/Wake button and the Home button at the same time for at least ten seconds, until the Apple logo appears. Release the Buttons.
More Info Here > iPhone User Guide (For iOS 6 Software)
I discovered that this wasn't the probelm. I tested my earphones from my iPhone 4s and they worked just fine, and then I tried my iPhone 5 earbuds again and I still didn't get the volume control to work on the remote. I finally pushed with much more force than usual, and then it would work once every 5 pushes. I'm guessing that this single set is defective. I'll see what I can do about getting a new set. Thank you!
I purchased the Klipsch S4i Rugged headphones from the apple store. The controls would not work AT ALL on my iPhone 4S using IOS 6.0.1. I tried them on a friend's iPhone 4S using IOS 6.0.0 and the controls worked fine. I traded the earphones for a different set, same problem. I guess it is a bug in 6.0.1.