Cutting up the piece of paper would not invalidate your licenced status itsself, in the sense that you could apply to the CAA for a replacement piece of paper without having to take all the exams and tests again. But it invalidates the piece of paper as PROOF of your licenced status - which is what the piece of paper is for - so you would have no easy way to demonstrate your status and priviledges as a licenced pilot to a person in authority.
In other words, of course you shouldn't cut it up!