What is the performance loss if you run Ubuntu desktop edition on a server machine?

As far as I know, there is no performance lost as far as overhead and whatnot. It mostly depends on what you have installed. You can turn desktop Ubuntu into server Ubuntu by installing the same security/monitoring/visualization programs. The server edition just comes with a better set of pre-installed packages suited to a secure, easily maintained server.

Either way, I would recommend NOT installing X server and a desktop environment (GNOME, KDE, etc). This reduces boot time and memory/CPU usage.

Almost all of the difference between Ubuntu Desktop and Ubuntu Server is in the default set of packages installed.

The only real code difference is in the kernel package - the linux-image-*-server packages have a slightly different kernel configuration to the desktop kernels. Such kernel options include enabling PAE mode (for accessing > 4GiB memory on 32bit systems) and changing the default pre-emption level (which prioritises CPU throughput over task latency).

These won't generally have a major performance impact.

Beyond the fact that an X server is running on the machine (and things like ubuntu-one-client once a user is logged in locally), there's really no difference nor performance impact.

There isn't a "server" version and a "desktop" version of Ubuntu where one magically limits the number of connections you can have to a machine (like some other "workstation" and "server" operating systems of years past).

The different install flavors are simply a different set of starting packages.