Thickness Measurement
Since you are not trying to determine the absolute height of the stack, but rather to detect a change, I'd suggest not measuring thickness at all. Nominal thickness of 20 lb paper is .004 inches. You could use essentially a microscope with a video camera, and adjust the height to provide maximum maximum contrast in the image. This is basically how cameras autofocus. You'd need a microscope lens to get the depth of field down to thousandths of an inch - regular camera lenses won't do the trick.

But, as I say, I'd recommend a different approach. Specifically, I'd go with weighing the stack, then detecting when the weight has decreased. A stack of 100 pieces of paper will decrease in weight by 1% when one sheet is removed, and that ought to be much easier to detect.