Iron on edge trim doesn't always stay put. Sometimes that is because cheap trims have very little glue and really should be glued with contact cement. But there are a few steps that will improve glue performance.
First, sand the edge so that it is smooth and flat. The cut from most saws leaves the edge a bit rough and the iron on glue doesn't reach a lot of wood fibers to stick.
Second, unless you have an exceptionally thick glue on the "iron-on" tape, you should put one coat of contact cement on the edge of the board as a primer. Let it dry for a full day. That will present a good adhesive surface for the tape.
Third, when you use the iron to apply the tape, you need to move slow enough to melt the glue but not so slow as to scorch the tape. You need to push hard enough to get a firm grip, but not so hard as to squeeze out all the glue. Did someone say this was an art and not a science?
Fourth, let it cool off completely before trying to trim it off or you will just slide the trim around and ruin the glue bond.
Fifth, cut a very slight bevel in the edges so that you cannot catch the trim as you run your fingers off the edge. (By the way, if you are laminating the surface as well, always do the trim first and have the laminate overhang and protect the trim.) The photo shows a tape trimmer which is simply a little carbide blade in an edge guide that trims the laminate down to a perfect bevel. A properly adjusted router can accomplish the same task.