Thank you for your help. I tried making config changes to Apache, but to no avail. It still seems determined to take over every IP address on the machine.
Since this is an Apache issue, I continued it here:
https://www.clearos.com/clearfoundation/social/community/can-t-get-apache-to-limit-itself-to-certain-ip-addresses

And this doc gives examples of listening on different IP's.

Hi

Can't you create Apache virtual hosts? Then you can have multiple domains on a single IP (on port 443 and 80)

https://www.digitalocean.com/community/tutorials/how-to-set-up-apache-virtual-hosts-on-centos-6

<VirtualHost *:80>

     ServerAdmin webmaster@example.com

     DocumentRoot /var/www/example.com/public_html

     ServerName www.example.com

     ServerAlias example.com

     ErrorLog /var/www/example.com/error.log

     CustomLog /var/www/example.com/requests.log

</VirtualHost>

<VirtualHost *:80>

     ServerAdmin webmaster@anotherdomain.com

     DocumentRoot /var/www/anotherdomain.com/public_html

     ServerName www.anotherdomain.com

     ServerAlias anotherdomain.com

     ErrorLog /var/wwwanotherdomain.com/error.log

     CustomLog /var/www/anotherdomain.com/requests.log

</VirtualHost>

I use a similar method to have a intranet site and a Wordpress Blog installation running on same system

I was just replying to your other post. Roundcubemail is not an app as such. It just forms another part of your web server (for me it is mydomain.com/mail, mydomain.com/roundcube and mydomain.com/roundcubemail, all aliased). I've no idea how you restrict it to a single IP, but it is probably possible. You probably have to research what you can put in the configuration files in /etc/httpd/conf.d/.

Thank you, Nick! I created Virtual Interfaces for the additional external IP addresses, and it appears to work.

A followup question: How can I restrict an application to a single IP/Virtual Interface? For example, let's say I want to install Roundcube to let users have webmail for the mail server on the machine. I want it to be accessible through a particular subdomain (webmail.mycompany.com) which points to an IP on this machine. But I already have an application that uses a web portal on this machine, and a different subdomain (timesheet.mycompany.com) points to a different IP. How do I make Application A available on Virtual Interface A and Roundcube available on Virtual Interface B, each on ports 80 and 443? If this isn't the right subforum to ask this in, where should I ask?

If both IP's are to be used by ClearOS you can create a virtual interface. If the one IP is going to be forwarded to a machine behind ClearOS, use the 1-to-1 NAT module.