Check Fight Club. I haven't read it, but I watched it, and in the movie, Ed Norton was just "The Narrator." Maybe Palahniuk refrained from naming him in the book. If that's the case, that's a must read just for the lesson.
It might be hard to do, but it sounds doable, and interesting besides. There are plenty of movies (Citizen Kane comes to mind) where the protagonist is the audience stand-in and thus the name is never mentioned, so novel-wise it could work in a similar way.
I can't speak for everyone, but I know I would notice if the main character I was reading about didn't have a name.
I am thinking about one struggle with not showing the character's name, because I think it's important that you know her name as the author, but just choose not to share it. It's hard to get to know someone who doesn't care enough to give you their name. The kind of person that doesn't tell you their name is either a mute, which would be a fun asset to work with, or an asshole, which, too, could be a fun thing to work with. I don't think many people enjoy the asshole protagonist, unless they're an asshole for a good reason. I think the question that would help make up my mind if I was thinking about doing this would be "Will it make my story better to have a nameless protagonist, or will it confuse the reader". Don't sacrifice readability for a gimmick. I just might play around with a mute protagonist... that sounds really fun right now... as I walked away I had a thought: People are "namers of things" and our brains are narrative machines. I don't think you can not give someone a name even if it "that guy with the crazy sideburns" that's a name of a kind. If the people in the story don't know her name, they will give her one. That'd be a good way to avoid her having a name, let other characters name her and never let the name(s) stick. you could force the reader to get to know the character over and over again before they realize that Charis, Lisa, and Tamra are the same person in different environments. People just started calling her those things at some point. That would force the character to be a wanderer of some kind, but that would be another way to leave a character nameless, just give them so many names it doesn't matter.
