Play Visual Voicemail In Background?

This may help you understand what is VM and who controls it. --> VoiceMail is a carrier-specific feature - Apple Community

By default, iOS restricts background audio playback for most apps to preserve battery life and prevent audio conflicts. This means that when you switch to another app, the audio from the previous app (including Visual Voicemail) might be paused.

 Play the voicemail using headphones and you might be able to switch to another app without pausing the playback. However, this might not work consistently.

That may be true, but Visual Voicemail isn't one which can. Visual Voicemail isn't really a streaming service, like other apps which do stream either music or video. It is a carrier specific feature.