I was watching someone play Call of Duty 4 recently and it struck me just how pretty that game was in Hardcore mode, Search and Destroy, when devoid of a HUD.
Granted, the game has top notch visuals as it is, but when there isn't a map, a directional indicator and various equipment, weapon displays and a crosshair, the game seems almost like something else altogether, almost cinematic. The only information displayed are the things of utmost importance being either the bombsites you must guard (or diffuse if it comes to that) or the location of the bomb and bombsites you must attack (or guard after being planted).
Everything else is moot, and I think that helps bring the game into focus. Those are your priorities.
I play on Hardcore mode exclusively, but this is something that didn't really strike me, I guess because as the one playing it my attention was always elsewhere (like not getting shot). Not having a HUD makes you pay much more attention to your surroundings, since it isn't obvious where your allies are, how many rounds you have left in your weapon, how many (if any) grenades you have. You have to keep track of all of that, and that makes the experience more engrossing, at least to me.
On the other hand, in a game like Crysis, I don't mind (and really need) the HUD. I need to keep track of my suit's energy and my health to know what mode to keep the suit in, when to change modes (health is low, switch suit from strength to armor, that kind of thing) and what condition my vehicles are in. The Battlefield series I have a similar need.
I actually used to play Desert Combat with my map always up and maximized, with transparency set to around 80% so when flying the entire map was overlayed on my screen so I could both fly and watch for points of interest. Knowing immediately what flag is in jeopardy, knowing where players spotted enemies, and looking at where a teammate disappears from the map and watching the kill announcements in the corner to determine what was there and if I could combat it. This is the polar opposite of how I play Call of Duty. So I guess it just depends on the game.