Should I peel off the top layer of vinyl in preparation for tiling?
Unfortunately, you NEED to remove or cover the old vinyl. If you attempt to tile over an unsecure base, you are wasting yout time and money because the tile will not stay down. In your case, if you go over the splitting vinyl, the vinyl will continue to separate and up will come the tiles.  If you go over the paper layer, the mortar/quickset will not stick properly. If you don't want to strip everything, which is a very thankless job, and your subfloor may not be the greatest either,  then screw down a layer of Hardi-backer. Hardy-backer or concrete board is the perfect base for your new tile.  Good Luck