How can I make the stock browser continue running while the screen is off? I am running a JavaScript compliance test in the Android stock browser which takes quite a long time. How can I prevent the browser from pausing when the screen goes off?

Meanwhile, I am using the "Stay Awake" debugging option, but I would rather avoid this because it may burn my screen.

I am looking for a solution that allows the screen to go off while preventing the device to go into sleep mode and thus allowing the browser to continue doing its computations.

Device: SGS3 running CM9
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.