Can a mobile browser continue to play audio when your phone is in your pocket? For example, YouTube stops playing when I put my iPhone to sleep.

That's a function of Youtube primarily delivering video content. A website designed to deliver audio content (e.g. Soundcloud) can keep playing just fine with the screen off.

Yes, it can. Start the video and put the iPhone to sleep. When you wake the iPhone up a moment later the music controls on the lock screen allow you to resume playing the audio. You can then turn the screen off and the audio will still play.

That's a good work around, but I would not want that to be the instructions I give to people wanting to listen to my audio stream from a website.

I do it all the time, so yes.