Tissue Paper has been used in tricks that rely on its unique quality to be greatly compressed.
The most famous of these is unquestionably "The Paper Hat Tear," invented by Cecil Lyle in the 1920s. In this famous trick, still performed by magicians hundreds of times a day all around the world, two pieces of tissue paper are displayed front and back. They are torn into many pieces and balled up. When the pieces are opened, they are seen to have turned into a paper hat with a flower on it. Many famous comedy magicians have used this, including Al Flosso and Bob Read.
The idea of using tissue paper for a trick where paper is torn and restored goes back even earlier. A remarkable version by T. Nelson Downs, where a spectator applies a postage stamp to the tissue paper prior to the tearing (and it's still there after the restoration) was published in MAGIC magazine years ago in an article discussing the connections between Downs and Charles Jordan.
The other major trick using tissue paper is "Red Ashes." Here, a strip of red tissue paper is lit by a match. The magician lets go and, as it burns, it floats upward. When the ash falls down, the magician catches it, squeezes it, and opens up the paper to reveal it's fully restored.