This XenForo 2.x addon sends a summary of XenForo logs to administrators via email.
Currently, only server error logs are implemented.
This runs using a XenForo cron task and can be configured to check for new log entries on a fixed interval (minimum 5 minutes). So you could have it send a...
Logs emails that are sent from your forum.
(Example of Email log link)
(Example of Email log page)
(Example of Options page)
(Example of User group permissions)
This add-on requires the use of the default email transport method.