The Metal Gear Solid games have allowed players to look around in first-person for quite a while, but to play the whole game like an FPS -- with convincing movement and decent controls? This looks like a job for modders.
The Metal Gear Solid V: Ground Zeroes first-person PC mod -- which you can see in action courtesy of Ben Blatcher -- straps the in-game camera to Naked Snake's* naked face instead of to the strange flying person who's always following Snake and panning down to his butt at inappropriate times.
It's still early, so there are issues with Snake's body turning invisible sometimes (not in an advantageous way) and certain animations not matching up, but it's pretty cool overall. I still think Ground Zeroes plays much, much better in third-person, but options are always neat.
You can grab the mod here. Installation involves mucking around in your game files a bit, so be sure to make a back-up of your game before diving in.
*I know he was technically only Naked Snake in Metal Gear Solid 3. The sacrifices I make for trite wordplay.