Hi,
I am not sure about the Tomcat and JSP part in your case, but I have seen similar issue with nginx. This is not a Docker issue, rather its something related to Virtualbox (I assume you are using boot2docker).
There is an open ticket for virtualbox for the same https://www.virtualbox.org/ticket/9069 which is 4 years old.
This was the fix with nginx - http://nginx.org/en/docs/http/ngx_http_core_module.html#sendfile
A similar issue was discussed here : Docker "Mount a volume on the Container" for Mac OS X kinda works
I am sure something similar will be there for tomcat.
Regards