After I cut a dichro strip, I wipe the edges carefully with a rubbing-alcohol soaked paper towel, this will help remove any residue.
Are you covering the dichro side with color or with clear? If clear, I start with running a 3mm rod down the edges with as soft a flame as I can manage, then switch to 8mm for covering the the rest. If color, the first and last stripes I lay down I put on at an angle to the edge so that it spills over a bit and covers the edge. This means that the strip will have thin black 'outlines' on the edges of the dichro strip (see pic below).
Either way, you need to cover the edges.
edit: 2nd pic shows this ribbon in use