You do need valid DNS services. But you don't need to provide DNS on the same server. And if you're not dealing with NAT, things can get easier.
The easiest approach available is to not run your own DNS services here. This assumes the OS X Server box is configured on a static IP address, but then that's something OS X Server needs/wants/ expects.
Use the DNS provided by your domain registrar, and your ISP. Or maybe on that Linux box, if that's publicly authoritative for the domain.
Enter the host name and the IP address into the public DNS services that you have configured for the domain, or that you have at your registrar or ISP, or on that Linux box.
You will need to have your ISP for the static IP configure a PTR record (reverse DNS) for the server, particularly if you're planning to run mail or related.
You can shut down the DNS server on the OS X Server box.
It serves no purpose here – if you already have public DNS services running for this host – and it's another piece to mantain.
As a test of this, you can shut down the DNS server, configure the OS X Server box to use the existing public DNS servers, and confirm DNS operations with sudo changeip -checkhostname or such.
As for the forward and reverse DNS records, anything that expects to use secure network connections will want the host names to match. I'd tend to expect network diagnostics to catch this error, too.