Backup, share, or save iPhone voicemail messages? Is there a way to export, save, share, or e-mail voicemail (not Voice Memos) from the iPhone?
I am guessing there are some under-the-hood ways to get at the files once you sync to a desktop, but is there any way to do it straight from the iPhone?
I have seen this response for jailbroken phones.
How can I backup and archive my iOS voicemail messages?
I believe there are applications that can extract the voice messages from your phone but they cost money and required the phone be plugged in to the computer.

The simplest way is to connect your phone to your computer with a 3.5mm audio cable. Use a simple audio recording app, such as Audacity, click record and play the voice message.