The Gaffers tape adhesive shouldn't degrade nylon over time as it is petroleum based - correct. Any concerns beyond the gunk? Yes I am too lazy to run to home depot.
I think it does degrade nylon over time, sure someone will correct me if I'm wrong. Why not wrap something like paper or plastic wrap around the links first, then tightly wrap the tape around that so there is no direct adhesive exposure to your gear?
Lowes or Home Depot, vinyl tubing by foot, I usually get a foot or two of 1" I think, but you should be able to eye ball in the store and the stuff is pretty cheap.