Remote Buddy doesn't do anything special that I could identify as being able to cause such an issue. It only opens a standard window and displays a standard view in it after all.

My best guess is that this - at its root - is a bug in the new Virtual Display feature of Lion Server that only Apple will be able to fix. Why?

I'm using a Mac mini server (the first version of the Mac Mini server) with Snow Leopard Server myself and have noticed something really, really odd: if I run the Mac Mini headless - with no display attached - and run standard software like VMWare Fusion, it doesn't display the VMWare Fusion icon correctly, the virtual machines' windows are generally just black and I see a lot of debug info being logged by the OS X graphics system: these log entries indicate the graphics processor (due to no display being attached) is in some kind of sleep state and optimizations inside the OS X graphics system that are written in OpenCL are failing due to it, leading to the display errors. The same happens when I start Time Machine. Remote Buddy, on the other hand, works like a charm, even without any display connected, probably because it only does simple drawing and - with one exception (that I'll explain below) - doesn't use any GPU operations.

Please check your system's logs for output along those lines. Please also check Prefs > Menu > Size & Appearance: have you selected any menu effect there? These effects use Quartz Composer compositions, which, in turn, may be using the GPU in ways the Virtual Display feature of Lion Server currently doesn't support. Just like VMWare Fusion and Time Machine on SL Server described above.

Best regards, 
Felix Schwarz