Is there a way to download or save voice recordings from messenger?
You can't do this on your iPhone itself, but you can on messenger.com on a Mac.


Open messenger.com in Safari and navigate to the voice recording.
Open the Web Inspector from the Develop menu (enable from Preferences → Advanced) and choose the Network tab.
Play the voice recording.
A new row will be added to the Network tab. Double-click the file and it will download.