I'm also having the same issue with a very latest Tomcat server (7.0.40). It goes non-responsive once for a couple of days.
To see open connections, you may use:
sudo netstat -tonp | grep jsvc | grep --regexp="127.0.0.1:443" --regexp="127.0.0.1:80" | grep CLOSE_WAIT
As mentioned in this post, you may use /proc/sys/net/ipv4/tcp_keepalive_time to view the values. The value seems to be in seconds and defaults to 7200 (i.e. 2 hours).
To change them, you need to edit /etc/sysctl.conf.
Open/create `/etc/sysctl.conf`
Add `net.ipv4.tcp_keepalive_time = 120` and save the file
Invoke `sysctl -p /etc/sysctl.conf`
Verify using `cat /proc/sys/net/ipv4/tcp_keepalive_time`