One should generally never remove a kosher tzitzis fringe from the kosher talis of a living person, out of respect for the mitzvah, unless his purpose in doing so is to use that fringe on a different garment. (Ed: We will learn about the custom of invalidating the tzitzis of a deceased individual prior to burial, iy”H.)
The remainder of this siman contains a complex and detailed discussion of garments that become torn in such a way that the tzitzis-fringes are separated and/or removed from them. In some cases, the tear can simply be sown up. In others, the tzitzis must first be removed and then tied on again after the repair. We will leave that discussion for another time. (Ed: Even though it is difficult for me to tear myself away from it!)
Note: Whenever one sews a tear in the “fringe-zone” of one of the corners, it is preferred that he not use a thread made of a material from which tzitzis could halachically be made on that particular garment - i.e. cotton thread to sew up a cotton garment. This is because there is a fear that he would decide to use a protruding piece of that thread as one of the ‘tzitzis’ strings, and it would not be kosher because it would have been attached to the garment not for the sake of the mitzvah, and switched after-the-fact!