I had same problem, but found a solution *despite* Apple's customer service.
Without access to the old credit card that was on file, I couldn't reset a password online, but the dialog said Apple would send further instructions by e-mail within a few days. There was no further response from Apple. Telephone customer service was absolutely useless -- they just had me walk thru the futile online reset process again. "What do I have to do, throw away the phone and set up a new one with a new user ID?" They didn't offer any alternative.
Later, while looking for info on resetting phone to factory settings without user ID password (which is reportedly impossible since iOS 7), I learned that the user ID password can be reset through FindMyPhone. It worked! So before you give up, see if you have activated FindMyPhone, and look for the password reset there. There might also be a solution if you have previously backed up your phone in iTunes, so don't give up hope.