Easy - use a proxy! If you like apache you will love apache mod_proxy for your purpose.
<VirtualHost *:80>
ServerName maindomain.com
ServerAlias *.maindomain.com
# insert document root and general settings for this domain here
# ...
ProxyPass /forum http://forumdomain.com
ProxyPassReverse /forum http://forumdomain.com
ProxyPassReverseCookieDomain forumdomain.com maindomain.com
</VirtualHost>
This configuration makes apache do a HTTP-request to your internal domain (forumdomain.com) without notifiying the users browser of the internal location. Your Forum will be accessable at http://*.yourdomain.com/forum. Cookies and headers the forum sents will get rewritten accordingly and Search-engines will not take notice of your backend-server.
You can read more about it at http://httpd.apache.org/docs/2.2/mod/mod_proxy.html
Should you need to rewrite reference sin your html (href, src ...) you might google on "mod_proxy_html".
A solution like this could of course be build with other intelligent proxyservers like squid as well. You can use it to map any content from "backend servers" to your public domain.
Make sure routing is OK or set up a host-entry for your internal domain (forumdomain) with a internet ip-addresse 192.168 ...
Enjoy your site and give feedback how worked out :)
p.s.: a "RewriteRule" directive can potentially do the same thing for you but the rdirect will be visible (and executed) by the client unless you specify the "P", foricng it to do an internal proxy request. If available I would prefer the mod_proxy though as it is more versatile and allows for more configuration.