At this point, only Batman and Spider-man continue with the old fashioned concept of needing a mask since their secret identities are so important for their characters. Nearly every live action movie superhero has a public identity or hasn't bothered to use a mask.
The X-men movies never had the characters in masks, all of the Avengers have public identities, Fantastic Four are maskless. Blade just needs a cool pair of shades.
Iron Man is the poster boy for superheroes with a public identity, he actually feeds off the attention.
I think the jury is out on whether Captain America is going to have a secret identity in the 21st Century. Once he gets acclimated to the modern world, he may want an ordinary life as Steve Rogers when he's not going into action as Captain America.
Sure, SHIELD and the Avengers will know who he is and how to find him, but he's not going to running around in the star-spangled suit 24/7, so the mask could help him preserve a degree of anonymity when he's out shopping for groceries.