Saving or sharing a voicemail used to be an impossible tasks, but not anymore. As long as your iPhone is running iOS 9, there's an easy way to save voicemails as voice memos, in a note, or to your Dropbox or Google Drive, and you can forward them via a message or email. You can even share them to other third-party apps if you want.
From within your Phone app, enter the Voicemail tab and tap on the voicemail you want to share.
Once selected, you should now see the share sheet icon—simply tap on it and choose which way you'll be sharing the file.
You can choose Mail, Messages, Notes, and Voice Memos, as well as third-party apps such as Slack, Pocket, Wunderlist, Trello, WhatsApp, Facebook Messenger, Google Drive, Dropbox, etc.