Are you using any 3rd party applications on your phone for voicemail? Try calling your own number from your cellphone and see if there are any saved messages you can delete.
I would then try leaving a test voicemail from another line to see if the visual voicemail is working on the iPhone. If not then you should perform a network reset of the iPhone by doing the following:
Reset your Network Settings. Tap Settings > General > Reset > Reset Network Settings. This also resets Wi-Fi networks and passwords, cellular settings, and VPN and APN settings that you’ve used before