Why can't I play music or podcast while playing Pokemon Go?

This has been updated and fixed as of iOS version 1.25.0 and Android version 0.55.0. You are now able to play music in the app without having it get paused/shut off.

From this article:

The update brings an exciting feature Android users' ways, which lets them listen to music while they are playing the game. Players are reporting that the updated app does not interrupt the music being played prior to the start of the game any longer.

I can confirm this behavior on iOS (the article seems to only mention Android). Music playing will no longer have its volume lowered or muted when using the updated app.

Yes, I ran into the same issue after the update and I was able to find a workaround.

If you are using iOS, activate Siri, then close out of it. Your audio levels will be back to normal afterwards.

Here is another workaround, in case the previous one doesn't work.

I'm not sure if this works for everyone (and it's not really a solution), but when I skip to the next track the volume goes back to normal. If I switch apps and reopen Pokemon it lowers the volume until I skip tracks again, but otherwise it stays fixed. I'm on iPhone with iOS 10.

It appears that there is a potential workaround for Android users as well. See this post:

Music is stopped each time you launch/reopen the game on Android. If your music player supports quick controls in the notification bar (e.g. Google Play, Amazon, Pandora), it can be restarted there. If they don't have a Play/Resume button there, it seems you're out of luck.

Additionally, it looks like this step may or may not be required:

you have to enable notifications for whatever music player you're using and then you can play it from the notification screen

If you use Poweramp on Android, you can disable this feature by going to Settings / Audio / Audio Focus and disable "Permanent Audio Focus Change". After that your Music will play on even if you switch to Pokemon Go.