ssh-ing into the machine, I discovered that the process coreservicesd was consuming 100% of one of the CPUs. I could kill this process and it would unhang the system, but then no icons would be displayed on any apps, and drag-and-drop would no longer work. The only way to fix this new mode was to reboot. Then clicking on the original folder would cause the hang again. I tried rebooting using the installer disk and ran Disk Utilities, only to find no errors on the disk. I tried rearranging the folder via Terminal to find that the app inside was the culprit. So, I moved it to the trash and deleted it. Upon reboot, the system would hang when logging in. coreservicesd was the culprit again, only now it would do it at login!
Some further poking around, using Find to find files that had changed in the last day, I found files in /Library/Caches had changed. Since coreservicesd seems to be responsible for providing icons and such to apps, this seemed a likely source. Renaming the folder and rebooting fixed the problem! So, it appears that files in /Library/Caches can get corrupted and cause coreservicesd to get wedged which freaks the whole system out, since it seems as though all apps use this service.
If this happens to you, check here. Also, for other issues that may arise one day, try finding files that have been modified in the last day to see what has changed.