"packing tape" can be made of various materials. The dielectric used in the Goertz foil inductors is 0.0015 inch thick polypropylene film. That's pretty thin... One might be inclined to just leave it stuck on the foil... if it is indeed "stuck" and then just insulate the other side. You can cuss-n-discuss "insulation" to death and get all kinds of opinions. To be honest... I've heard "good sounding" speaker wire with various types of insulation. I think the "design" is at least as important as the material. And I also believe that the thinner the insulations is, the less difference it makes to the conductor what its made out of (within reason of course).
Goertz's other foils can be ordered with various polyester or kapton film insulation. Goertz foil speaker cable is advertised as being insulated with very thin films of Teflon/Polyester Terepthalate/Polycarbonate. Polyester Terepthalate is commonly referred to as "PET". Expandable insulating sleeving, i.e. TechFlex is made of this kind of material.
Goertz is the hobby division of Bridgeport Magnetics, who makes very high-tech transformers for the medical industry for one. I tend to think they know what they're doing......
I, for one, would not get too excited about the very thin insulating film used on the inductor foils... if it's reasonably well stuck to the foil - I'd just leave it on there and go with it... see if you like the way they sound. Path of least resistance, least expense - first.
I like "gaffers tape"... but you have to get "real" gaffers tape, which is simply an adhesive backed cotton cloth. 3M makes some.. you can buy at parts-express. It is relatively expensive as "tape" goes... but it's the real deal and I've had a good experience with it so far. It's not pretty to look at, but you can put some PET (TechFlex) sleeving over it if you want to pretty it up. There are many who will tell you that TechFlex sleeving mucks up the sound of wire/conductors... and that it attracts static electricity..... who knows....