Most of the components of the GameEngine are licensed under the LGPL, so if you want to link them statically you have to share your project's code, if you link dynamically you only have to provide your changes to those components. The parts not licensed under LGPL but GPL, are probably only not adjusted to the LGPL, but that's something I think I can't decide, since I'm not working at the university anymore. And concerning the FACS component I'm not sure, whether it will be adjusted or not, maybe you can ask korty
If you develop a game and do not plan to release it, I don't think you have to care about all those license issues. If you plan to release it, you don't have to ship your game with the source code, you only have to mention that you are using LGPL components and which one and you have to provide a possibility to get the code for these. If you haven't changed anything at the LGPL components, it should be enough to mention a link to the GameEngine project website. In case you have changed something you may also need to provide a patch for the original LGPL components or the complete changed source code of this component.