@worldsendless your best bet is probably to disassemble it, slip some heat shrink tubing on it, reassemble it, and hit it with a hair dryer. Won't come off, doesn't have seams to betray the repair, pretty abrasion resistant.

If you must have tape, a black cloth hockey tape is probably the sweet spot between inconspicuous and durable? Just make sure you press the edges down well all along the length of the tape, or the adhesive might grab a chunk of hair when you go to remove the headphones.