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.
The possibility of conflict between application being installed and already running one is very small. These days most applications are based around concept of sandbox and they don't need to step out of it during installation process (or normal usage).

But lets suppose while installing some software, a change needs to be made to a file that is very likely to be already open by another application. You have two choices here:

1. check if the file is open/locked, figure out which process is holding a lock, display a nice dialog asking to close that specific application
2. just ask the person to close all the applications and you're on a safe side with no extra effort

Sadly, the second option gets picked up most of the time.