I agree 100%. Someone with very pale skin naturally, who does not tan with the sun...especially a natural redhead, looks horrible with a tan. (Check my avatar...

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My advice, wear colors that suit your pale skin/red hair. Black and green are my favorites. A redhead with pale skin looks great wearing black capris, black sandals, and some chic black sunglasses...or a retro/pinup style bathing suit.
I managed a hair salon for a long time...and the same thing goes for natural blondes (with light eyebrows/lashes) who want to change their hair to black, to look punk/goth. They simply do not have the coloring to pull it off...and it looks horrible.