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: 
You can't overwrite a DLL or EXE that is in use by another app. The better written prompts say "look, either close stuff that is using things I want to update (and gives you a list) or continue the install and we'll have to reboot after to replace the DLL/EXE then." The lazy ones just blindly tell you to close every single thing you have running, just in case. 

I hate rebooting. I hate closing stuff I'm using. Sometimes I just keep right on rolling and don't close anything that's running. I have a bit of a guess if things might share a DLL, so for example if I am updating Word I'll agree to close Outlook, but not Foxit Reader, I know they're probably independent. I pay more attention to non-lazy prompts (that tell me what the conflict involves) than to lazy ones, which may as well say "may contain peanuts" for all the thought that has gone into them.
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.