How to prevent Chrome tabs or any kind of app from sleeping when in background? I want to listen to things on Youtube (app) or on the web (Chrome) while doing other things on my Asus Transformer. Is there any way to prevent background Chrome tabs or background Apps from falling asleep?
No, it's designed like that by the app's developers, so you can't.

Though for YouTube, if you're rooted, there is a Xposed Module called Green YouTube that have an option to play music from YouTube in background (but it doesn't work on my phone).