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: 
A lot of times that message is just inserted as default text for the installation system being used. 9 out of 10 times it can be ignored. Its just that 10th time you have to worry about :-) imo, in addition to the technical reasons DevArt summarized nicely, it was probably put in there originally so that end users dont go screaming to vendors about lost work if the installer crashes Windows and causes them to lose important spreadsheets and documents that were being worked on - *a not too uncommon occurrance back in the day* when installing software.
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.