It seems that Apple's 'Visual Voicemail' is all very well but when you delete a message it doesn't actually get deleted - it gets put in a "recycle bin" taking up space and slowing the phone down or 'freezing' it (and hanging around when you though they had gone). This seems to be a particular problem on iPhone 4's running iOS7, though can also affect newer iPhones such as the 4S, 5, and 5S.
To fix this open the Phone app and select voicemail, then scroll right to the bottom of the list of voicemails where it says 'Deleted Messages'. Tap on that and you'll see all your old voicemails - to get rid of them tap on 'Clear All' at the top right and they're finally wiped away. You might also want to delete some of you existing voicemail messages to gain some additional space, and generally speed up voicemail and the Phone app a bit more.
If you're having trouble with iPhone freezes, or it going slow, when attempting to delete all those voicemail messages then try switching on Airplane Mode and deleting them all, then switching it back off again when you're done.