My Note has a tick box under phone/call settings/ringtones and keypad tones/dialing keypad tone
it was turned off (in the last update?) when I turned it back on it worked regardless of media volume.
Keypad tones and pressing numbers to input a voicemail password are two different things. I believe it actually has more to do with the network. The server, or whatever it is that stores/runs voicemail is what need to register the keys you press, which makes this a strange issue.
You can turn keypad tones completely off and should still be able to enter numbers while on an automated system.
Have you tried contacting vzw support? I would lean more towards thinking that this is an issue on their end. Especially if it was working previously and you haven't changed anything in your call settings.