Try to enable Dxwnd + GDI Mode to see if that makes any difference. Window mode might be an option too if it still doesn't work
GDI Mode is already on. The problem isn't exactly that I can't alt-tab. It's more that when I do alt-tab, the black screen of the game overpowers the desktop, so I can't go to the task manager to close the program. But I'll definitely try windowed mode.
EDIT: Windowed mode got tried, and it failed. The exact same issue appeared, and the desktop got REALLY weird. The Dune 2000 application itself seemed to flicker in and out of existence at times, and the big black box was staring me right in the face, still overpowering everything else on the screen.