While reading through the GPL FAQ I found some relevant points to the question, however it is still unclear to me if this fully applies to a public gameserver.
I'll try to keep it short:
There is a gameserver A, fully available, including source code, published under the GPL. There is a modified gameserver B, based on server A, not available as binary nor as source code. Maintainer of modified server B however does grant access to his modified server as a public service, i.e. to play the game, obviously.
From the GPL FAQ:
Does the GPL require that source code of modified versions be posted to the public?
The GPL does not require you to release your modified version, or any part of it. You are free to make modifications and use them privately, without ever releasing them. This applies to organizations (including companies), too; an organization can make a modified version and use it internally without ever releasing it outside the organization.
But if you release the modified version to the public in some way, the GPL requires you to make the modified source code available to the program's users, under the GPL.
Thus, the GPL gives permission to release the modified program in certain ways, and not in other ways; but the decision of whether to release it is up to you.
So to me it is not obvious what does apply here.
On one hand the "modified version" is clearly "release[d] to the public in some way" to the "program's users" (i.e. the players), however on the other hand it is kept as an "internal" tool.
As I understand it, "internal" in this context suggests, that the software can be used without publishing, if the public has nothing to do with the software itself.
But the server, and the providing of the public service in this case is not just part of something unrelated but the very core and reason of existance of the whole public project.