Question: What kind of problems arise from installing a new program while other programs are running? I'd imagine the GAC solved a lot of the old problems of shared dll's being locked. So what kind of issues still arise? I can see issues with upgrading a single program for sure, but outside of this its hard to see why apps would stomp on eachother's toe's.

Answer: 
maybe the installer is just overly paranoid?
in all honesty, I have never ever closed all programs when an installer asked me to do so and never ever got into trouble. Might just be luck though.
it's hedging. i never bother, and have had no trouble. the only exception is installing a plugin when the targeted app (IE, FireFox, Visual Studio, etc) is still open