I have a customer with a 7200 using a ITP-5121D phone. They currently have this phone in their office, connected to the same LAN as the 7200. If I set the phone to use the private IP of the 7200, everything works fine.
If I connect the 7200 to a Cisco RVS4000 router, which has a DMZ set up passing all traffic to the 7200 (router has a static public IP) and the router's firewall turned off, and then change the phone to use the public IP, I can connect to the 7200, the phone correctly identifies the user and extension number, but I can't hear anything in or out.
Any ideas?

As far as I know you can only have 1 IP address in the DMZ. If so then no traffic can hit the OAS/MGI hence no sound. Port forwarding or VPN connection would be the way to go.

Is there more than one IP address for this thing? I could only determine one of them. How can I find out what the other one is supposed to be?

Is it a 7200 or a 7200-s and what kind of cards are in the system?

I don't know. How can I tell?

I really don't mean to sound like an idiot, but i'm a computer guy who tries to help with phone support. I really want to help this customer out, so any assistance would be greatly appreciated!

If you have a OAS/MGI card you will need to forward the ports to this card's ip address.
also need to check the programming to ensure the public ip address is programmed in

Check in the MMC840 for the IP Type to be public or public/firewall.

Also codec G.729a uses more compression on slower links.

If the system uses an MGI card instead of an embedded then you need to port forward UDP 30000 to 30031 to the MGI ip address and the other documented ports to the phone system ip address.

Hope this helps.