Rust on a vise head usually comes from the same place it does on blued or similarly treated metals, from perspiration and the oils & acids in your skin.
At least in my case, if I handle a raw steel, carbon, blued or similarly finished piece of steel and don't wipe it down with something other than a dry rag, I leave fingerprints that will rust.
Even the aluminum parts on my Renzetti Master have darkened slightly over the years as aluminum will do after contact with acids or alkaline substances. For that reason I wipe the entire vise head down after tying sessions.
If you don't have hands like mine that can rust stainless steel, do what ever that floats your boat with your vise.
If you are in the habit of wiping down a blued firearm or a carbon steel knife blade after handling, you probably should do the same with your vise head. In my case my vises are cost more than some of the knives & firearms I own so I don't treat the vises any differently.
Slipping one of
these over your vise head between sessions isn't a bad idea either.