There's a simple solution for those who are tired of seeing iconic pictures: Stop looking. I'd also suggest that if you're tired of iconic pictures and actually do stop looking you'll also stop looking when you're out with a camera. As HCB famously said, "Photographing is nothing. Looking is everything."
The possibility is, particularly among newer photographers, that the more inundated they are by particular images, the more it may influence their personal vision (the operative word here being "inundated"). That said, I introduce my beginning students to the best of the "iconic" photographs (of course I know that is redundant, but it makes my point), so that they may experience what I, at least, consider to be the best of the best... equal parts of Haas and Henri, Weston and Winogrand. Best of the best is exactly why these photographs have become iconic. Almost without exception they are compositional masterpieces (I may need to exclude Winogrand from that comment, but even his works have something to teach).
Just barely tangential: So far as I know, no image in photography begins to approach the level of inundation I experienced on my first visit to Vienna, where the streets are literally paved with Klimt's "The Kiss"... posters, flags, imitation paintings, glasses, cups, saucers, complete dish sets, t-shirts, tablecloths, dresses, neckties, _______ (just fill in the blank and they have done it), and it is almost impossible to escape.