I've been playing with my Gear Live for a couple of days, so here's what I can tell :
- The screen is "Always On" by default, which means it will get dimmer and switch to a black background on most watch faces (and for AMOLED, black means "off").
- You can set it to turn off completely.
- In order to dim it (or turn it off depending on the settings) you can either wait, cover the screen with your hand, or use the physical button... The button is not really practical and seems to serve absolutely no purpose since you can use the screen.
- I haven't noticed any burn-in at all yet
- I think I noticed that from time to time the facewatch (I'm using the one with the digital hour in the center and analog minutes, I think it's called "digitalog" ) switches a few pixels to the right or to the left. My guess is it is to prevent burn in, I know most plasma TVs do that too.
- I had a Galaxy S2 for several years before switching to a Nexus 5, and yes, the toip of the screen where the notification bar usually is was burnt-in (or rather I think the whole screen was burnt except for this black bar). It was noticeable on full screen apps but never bothered me. Even if it happens on the Gear Live I can't see it bothering me... It's not like I'll be watching movies on it...
So my guess is you should be fine... But if you're really worried you can always disable "Always On".