Honestly, that sounds like a carrier issue. If your phone is in Airplane Mode, cellular calls aren't reaching it at all and any error messages are being generated by the carrier's switch.
You might want to consider logging into your voicemail the old-fashioned way (on my carrier, you dial *86 and send. Yours may be different) and see if there are voicemails in the deleted section. Also, on the iPhone, go to Phone>Voicemail>Deleted and delete anything hanging around there. That may prompt the deletion of things on the server.
If all that fails, contact your carrier for assistance.