Our current approach to warning about other Tomcat webapps is prone to false positives, including:
Deploy LabKey to the root context in Tomcat.
Start up the server in production mode.
Create a /docs project.
Bounce Tomcat (the check is done during startup and the results are cached).
Log in as a site admin.
You get a warning that says "This server appears to be running with one or more default Tomcat web applications that should be removed. These may include 'examples', 'docs', 'manager', and 'host-manager'"
As we discussed in chat, we should consider switching to an MBean-based approach for checking for other webapps. Possible reference code: