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.
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