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I suppose you mean should you shine up an old axe head so it looks brand new, or keep the patina? I think most would agree that's a purely aesthetic thing, and therefore very subjective. Some guys/collectors love the look of the original patina contrasted with the shiny, newly sharpened edge, while others go over the entire piece and take off all tarnish.
If it's a working axe, you do want the cutting bevels and cheeks to be clean to keep friction down, but that doesn't mean you can't keep the patina. Look around the forum at some of the project axes guys have rehung and sharpened, you'll see great variety.
John