iPhone voicemail won't play
The issue turned out to be something about my iPhone audio. A little later in the day I discovered that music wasn't playing. (Oddly, I was still getting sound effects from a game.) The music app would advance the progress bar on a song but produce no sound.

With that additional data, I rebooted the phone (full power down and power back up) and music would play and so would the voicemail. So whatever it was, it was software (because the power cycle cleared it).