I don't think it would work. The cookie cutter edges are sharp, but not sharp the way die cutting edges are sharp. Also, I'm sure the material used for metal cookie cutters is softer than that used in dies. The machines use a mat between the fabric and roller too.
Are you thinking in terms of fusible applique shapes? Your method might work well enough to create indentations in the fabric which you could then cut out with scissors. Otherwise you can draw the shapes on freezer paper, iron the freezer paper to the right side of your fabric, iron fusible to the wrong side of the fabric, then cut with scissors along the freezer paper shapes and peel off the freezer paper.