I think one must differentiate between Arena/Map/Multiplayer Only based FPS vs. RPG-FPS (Deus Ex/Destiny etc) or Story-driven-FPS (Metro:2033 / Wolfenstein / Bioshock? / Doom?). I won't be burning out of the latter easily, simply because alot of it is about the story and experience.
That said, on the Arena/Multiplayer FPS side, again one has to make a difference between an FPS that is an "annual release" vs. an ever expansive/enhancing FPS.
CounterStrike is the latter , this is why everyone still plays it. It's 1 game that has changed over time. CoD/Battlefield is a new game every year, it is a continuous spend and also fragments the community, only a % of people actually move to the latest version , alot would stick to say Battlefield 4 instead of Hardline.
Imagine what CoD or Battlefield would be , if it followed the CS trend of community modding and expansions vs. stand alone re-releases.
For some reason i think i feel burned out having to buy a new game, start at the bottom, relearn something that is not new at all.
I have more hopes for something like Overwatch, since i don't foresee Blizzard making a new version every year, i expect them to enhance/add to it for a few years before releasing a new version....so lets say 3 years of the same Overwatch game sounds less "burnout prone" to me, since i can jump in at any time, play a bit, maybe buy a expansion here and there, jump out etc etc. Whereas if you had to buy a new CoD , you'd feel guilty for only playing 10 hours of it before getting sick of it...
...so i think people burn out on CoD/Battlefield because they just paid $60 for a game with different maps (from the one you have been playing up to that point) and feel "forced" now to get your money's worth so you play it more than you would if it was just a free expansion.
The same applies to something like Assassin's Creed...imagine if they just expanded and upgraded AC 1 or 2 and let the community enhance it for 3-4 years before releasing something that is new enough to warrant a new "platform" to build on.