One trick to get melamine tape to stick to the edge of a board is to prime the edge with contact cement and let it completely cure overnight or better still for several days before ironing on the tape. This will provide a very good base for the tape adhesive to stick to.
When it is time to trim off the tape there are basically two manual ways to do this without having to resort to power tools like those used for trimming the much thicker Arobrite or Formica.
Take a chisel at least an inch wide. Lay the chisel flat on the board and push into the tape. As soon as you are moving just a little bit up the edge, twist the chisel so that it continues to sit flat on the surface but it is pointing into the board. This will avoid pushing the tape off while cutting.
There are special razor cutters that will trim both sides of the board at the same time. It is as simple as squeezing the gadget onto the board and running forward. These are commonly available wherever they sell the melamine tape.
Before you quit, just pass a piece of sandpaper or a smooth file on an angle over the edge to "break" the sharpness of the trim job that will feel uncomfortable to the touch.