Doom? GoldenEye? COD Modern Warfare?
Doom would definitely be a contender for most influential FPS ever. While it builds heavily off of its predecessor Wolfenstein 3D, unlike Wolfenstein it was hugely popular, and helped make the genre mainstream and set the template for almost every FPS for the next 8 years until Halo CE came along and set the new standard.
GoldenEye 007? While it was the first hugely successful FPS on consoles, it was not the first FPS to be on a console, either. Doom 64, Hexen, and Turok were all released before GoldenEye, showing that consoles were feasible on consoles. Also, GoldenEye's approach to campaign design never became a standard, and was only ever emulated by its pseudo-sequel Perfect Dark. GoldenEye did however help popularize local split-screen multiplayer for FPSs, but that's a relatively minor contribution compared to Doom and Halo.
Modern Warfare? Wasn't even the first game, or even the first popular game, to have the same core gameplay. Yes, it did popularize killstreaks, perks, and rank-based progression systems in the genre, but I would once again that it does not make it more influential than Doom or Halo.