No, the FPS genre is not dying. I think we focus too much on AAA studios and publishers to be the only ones to deliver a high-quality product with a new/fresh experience when in most cases they cannot do it simply because AAA game budgets are too huge with overbloated teams that it is impossible to coordinate experimentation. To those looking for new experiences, look around. There are many developers who are making the games AAA cannot deliver. There is stuff like ARMA 3, Insurgency, Reflex, Shadow Warrior 2, Overwatch coming out in the future.
Remeber that FPS have gone through a few trends over the past two decades. One game did something successful and everybody else needs to emulate their success rather than try something new.
We had the Sci-Fi shooters with DOOM, Halo, Unreal, Half-Life, Timesplitters, Perfect Dark, etc that dominated the late 90s and early 2000s. Of course the game industry was still relatively young back then where all it took was 15+ dedicated individuals to create something groundbreaking and experimentation was the norm.
We then had World War II shooters with games like Battlefield 1942, Call of Duty, Medal of Honor, Brothers In Arms, and Day of Defeat being the norm from 2002 to 2008.
We have the infamous Modern Military First Person shooters with games that were once WW2 shooters(COD, MOH, BF), Homefront, Warface, Rainbow Six: Vegas, Ghost Recon, Operation Flashpoint, etc and it plagued the 7th generation because COD4: MW was a smashing success.