Well, I believe the only possible way you could do this is if you Mod the .xml code, but a more experienced modder should be the one to give the details on how to do that.

Even then, I'm not sure you can turn an entire screen off - the game might get a little buggy. You could always edit what they say, so instead of telling you what tech they're researching, they could just pop-up and say something totally random, like: "So....how's your day been? Any new conquests lately?"

TBQH, I think my version would creep me out more then the original. Having an AI randomly ask how your day is going would be a bit......bizarre.

I must admit, I'm a bit curious about one thing. What is it about this pop-up that bugs you enough to want it gone?

I've never seen the AI change tech goals that fast.  Not even when the plague or disease mega event occurs.  Generally it takes them a minimum of 5-8 turns to "react" to anything, as they prioritize finishing their current tree first.

On the other hand, if they keep calling you up to tell you "Hey, we just finished Lasers!", "Hey, we just finished Laser Mark IIs!", etc, then that's not terribly unusual.  Some AIs are more chatty than others.

I'd agree it's somewhat annoying, though.