What realistic issues can occur from shrinking the log files on my database (either through the SSMS UI or issuing a DBCC ShrinkFile command)?
If you're making regular backups of the log file or are running in SIMPLE recovery mode, the log file should mostly consist of free space; shrinking it will just truncate the free space from the log file.  This is usually safe to do, but you need to realize that the log file will likely just grow back to the previous size....when this happens the log will auto-grow, which can cause a performance hit every time it happens.

The only real reason for shrinking it is if you're low on disk space, if the log is filling up the drive you should consider adding additional disk space as a long term solution.