How can I make the stock browser continue running while the screen is off?
I don't believe there is any way to get it to keep running while the display is off, though I could be wrong.

What you could do is turn the screen brightness down all the way, and then use an app like Screen Filter to make the screen even darker.  At it's minimum setting, this will almost have the effect of turning the screen off, while letting everything still run.

I'm not sure modern screens even still suffer from burn in though.  Some do suffer from ghosting, but that goes away fairly quickly.  Burn in like on old CRT monitors was permanent.