are you allowed to play background music during your runs?

Short answer: If your submitted time is not competitive, it's fine.

Long answer: If you play copyrighted music during your runs, it's likely that your Twitch video on demand will become completely muted. If Youtube, the same could happen. This is because broadcasting copyrighted music while playing an unrelated video game doesn't qualify as "Fair Use". Any run that is submitted without intact game audio is less authentic than with game audio. That also includes having Free Use or Public Domain music drowning out the game audio. High authenticity matters to the mods when you're submitting competitive times.

^^^^ This.

There are ways to set up OBS so that you can stream your music, but you can locally record just your game audio. I don't personally know how to do this, but I'm certain it's possible. That way, you can play with music, and then upload your vods with pure game audio and submit those later.

If you don't care about streaming the music you play (in other words, it's for you, not your viewers), you could just listen in headphones, or mute the source in OBS and make sure your mic isn't picking it up.