What OS (Win or unix)? Do you know if the application doing the logging uses flock? Do you have any control over the code of application doing the logging?
One more question. Does the application keep the log file open, or does it close and reopen it. This can be answered using strace or your system's equivalent.
If it does close and reopen them, you can rename the file. Once the application creates a new log file, you can do whatever you want to the renamed one.