I’ve heard this myth too. Not entirely sure how much there is to it, but supposedly paper/tissue shouldn’t be your first choice for cleaning your face. Whether it’s because of the texture or “absorbancy” (the world of cosmetics are always inventing new words, so I’ll take the liberty..), I’m not sure, but it’s supposed to dry out your skin more than cotton or fabric cloth.
Personally I use cotton pads or the facial wipes (they’re by far the quickest and easiest option) to clean my face – and I’m frequently estimated to be younger than I am ;) of course, behavior may have something to do with that….