What do you mean by "FPS" exactly? Some people only use a very narrow definition for that term.

For me, I include games like Fallout 3/NV, Deus Ex, Far Cry, and Borderlands, all of which have a lot more to them than simple point-and-shoot. The new Thief should be very entertaining, as well, and Destiny is on the horizon.

If you really like FPP then there is a good amount of variety out there. If you just want military style multi-player, though, then I can't help you. Personally, I love FPS games but I don't care much about twitch combat.

Release less frequently.

Just go back to the roots. Make health packs and more than 2 weapons at a time part of gameplay. Don't have limited sprint and don't go for realism. In Doom you ran around punching demons in the face or blasting them with a shotgun that sounded like "god slamming his car door" (Zero Punctuation). Don't bother with cinema let the moments happen organically. Resistance 3 is my favorite FPS ever because it had a good storyline but didn't let that get in the way of putting stuff in that made no sense like upgrading weapons with use and cooky freeze and germ guns and health packs and carrying around 10 weapons at a time.

Also how about instead of all this class based, get all your weapons when you spawn gameplay (which is fine and should still be game modes) we get some old fashioned MP matches that give you a pistol to start and have weapon pickups?

Just get away from military fps and makes games like this year's Bioshock Infinite and Metro The Last Light.

Another way to freshen up the fps genre is mix it other genres. FPS/RTS havent been done much.

You suggest to reinvent a genre by adding something that's similar to another fps? 

logic

TF2

Need more FPS like Dust 514, Destiny, and Planetside 2. If the concept of those games were to be combined it would be awesome.

cross-platform play, call for orbital strikes from someone in another game, and the crisp graphics of killzone: shadowfall. This would be an FPS I'd like to play.

Even thrid person shooters like Metal Gear Online is a lot of fun. Damn I miss that game..

Well, they could go full MMO like Eve Online, where other players organise missions for other players to participate in until all out war breaks out for resource x, y or z.

Or you could play something other than military FPS like Bioshock Infinite, Dishonored, Deus Ex etc.