Privacy and Server Logs
While it may not be directly related to the security of your server, the privacy of those who access your site is something else to be mindful of and take steps to guard. If you use a commercial server, which also encrypts the logs, you're all set. Otherwise, be careful with the permissions on the logs directory, as previously mentioned. Also, find yourself a good log-analysis tool and use it frequently, then truncate the logfiles. It's inappropriate to use the logfiles for any purpose, other than overall traffic analysis, error-checking, and usage statistics in the overall sense.