The visual voicemail area of the Phone app now has a share button! You can send the audio of the message to a ton of apps—Email, Notes, Messages or Facebook Messenger (to send to others!), Voice Memos (to edit), even biz apps like Basecamp and Slack. Or shoot it to Dropbox to save in the cloud and on your desktop as an .m4a (MPEG 4) audio file—we call that a "forever blackmail." It's also a quick way to make voice memos to yourself when you can't access your phone; simply transfer your own message to other apps.
Open Voicemail on your iPhone, tap on the caller, tap on the Share icon, then tap to Share on the App of your choice.