I am sorry for your loss.
Let me try to help you. If your iPhone is not jailbroken, the easiest way is to connect a 3.5mm audio cable on your iPhone's headphone jack and your computer's mic or line-in jack. Download a software like Audacity, click on Record and play the voicemail on the iPhone. This is what was used to be called line recording.
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Why is something which should be so simple be so difficult.
A friend recently asked me if it is possible to download a prior VM from his Dad, who passed away last week???
I said, "I am sure this is quite simple on an iPhone" Right!
To my chagrin is it NOT SIMPLE :-(
What do I tell my friend who just lost his Dad???
I'm going to try bontistic's solution (thank you!), but I can't help but agree with Fred's sentiments. It's so surprising to me that there isn't a simpler solution! Perhaps I'm being naive or ignorant - to be fair I'm more of a newb than an expert when it comes to ios in general and developing in particular - but it seems like there could/should be a way for Apple to create a tab in iTunes (when device is connected, of course) showing voicemail files as data which can be simply saved to the computer (a la the way Apps can be shared between the device and one's iTunes "Apps" folder).
Sorry for your loss. If you still need help, try either of the following:
This Mac app will read your iPhone, extract all data to catagories, including voice mails. You can then copy them elsewhere on your computer or load them into iTunes.
Allows you to examine and extract anything form your iPhone backup, including voice mails.