Voicemail? Yeah, it's still a thing.
And sometimes a very valuable thing, like possible evidence in a court case, or even the last message received from a loved one. (Sad to think about, but it happens all the time. In fact, there's a Web service, VMSave, devoted to-- though I think that's hardly as meaningful as a message left for you.)
Anyway, whatever your reasons, Apple's visual-voicemail feature allows you to save messages to a variety of destinations. Here's how to find and use that option:
Step 1: Open the iPhone's Phone app, then tap Voicemail
Step 2: Tap any message to expand it, then tap the Share icon (represented by a square with an arrow pointing up).
Step 3: Now you can send the voicemail (as an M4A audio file) via email or text message, or even copy to the iOS Notes or Voice Memos app. You can also tap the More icon (the one in the lower row) and enable Save to Dropbox, which will then add that destination to the Share options.
It's worth noting that M4A, the extension for audio files encoded in the AAC format, should be playable on most PCs and mobile devices. However, if you prefer MP3 or another format, it's easy enough to convert these audio files. CloudConvert, for example, is a free Web-based tools that can get the job done.