Lead DOESN'T RUST, it's non ferrous(meaning it doesn't contain Fe2+ which is IRON). It can be oxidized when water or some sort of chemical strips electrons from it, but usually for things like lead pipes it happens at a very low rate.
The brown/ dark gray color is just discolorment from a source. Could be from sweat, could be from the glue under the grip.
I would suggest always covering lead tape with some sort of clear tape or something else, because its still lead and very poisonous to humans.