How to remove a rectangular region at the centre
You can use the drawbox filter to umm. draw a box on top of the video.

Basic syntax is 

ffmpeg -i in.mp4 -vf "drawbox=X:Y:M-X:N-Y:c=black:t=fill" -c:a copy out.mp4


where X, Y are replaced with top-left co-ordinates of box. M-X and N-Y are width and height of the box. The value fill for thickness will make a solid box. This value was very recently changed, so if you have an older version of ffmpeg, use max instead.