FireShot stores license information in the registry and in the settings file.  If your account does not have permission to write to the registry or the local settings file, the next time the software is started it will revert to an evaluation copy.  The workaround is to log in with administrator rights, install and input your key.

In Windows Vista or 7 you may have to exit the program completely, then right click on the shortcut icon and select Run as Administrator.  Then put your key in when prompted.