I searched high and low across the internets and even went to the second page of Google to look for a solution but it seems like nobody else has ever even had this problem.
Much of the time (but not always) when going fullscreen either via the F11 key, or by clicking a fullscreen button on a flash video, rather than actually going fullscreen, the firefox window just completely disappears. There is still a firefox icon highlighted down in my windows taskbar, so the browser is still running, but clicking it (as if to restore the window) doesn't do anything. The only thing I can do at that point is right-click the icon in the taskbar and choose "close."
This will always happen, for example, when watching a YouTube video embedded on an external site and clicking the fullscreen button on the video. However, it will never happen if I'm actually watching the video on YouTube. It is just crazy like that. It will also sometimes happen when pressing F11, but not all the time. It is seemingly random with F11. It happens on many other flash video sites (you guys know which ones) but not all of them.
I'm running Windows 7 x64 with an aero theme and my video drivers are up-to-date.
If anyone can actually post a solution that results in me fixing this, I'll note your answer, come back in two days and bounty the question and then just give it to you.