How long will it be before COD burns out, or Activison pulls the plug on the franchise?

I think that Activision will stop milking the franchise when it stops selling, and it will never stop selling.

I think it will head into decline phase starting next year, but won't hit an actual burn out till much later. 

I am a huge fan of the COD series, however, my interest in the next installment has diminished greatly.  I think I might just be burnt out with the series myself.

Never. The Dev teams won't let it happen. 

What happens to the FPS genre and videogames in general, if Call of Duty burns out?  Is the FPS genre heading the way of music games, if Call of Duty ends up crashing and burning?  What else will it fall back onto?

The next installment, this November seems to be the one that will downtrend in comparison with the rapid increase in sales from COD4 onwards. The happenings at the IW have played the franchise into questionable hands, even if the game is good it will most likely be compared to the works of IW and Treyarch. After that I think a few more games until it starts downtrending. I found Black Ops to be a hybrid of MW2 and COD4, the progression in this genre is rapidly deteriorating and the multiplayer is becoming more of the same, I don't think I know many people who buy COD just for the single player.

I'm not sure. In the past 40 some-odd years of videogaming, people still haven't gotten tired of shooting thing. We've only been able to do it from the first-person perspective for about half the time.

I'm guessing by burn out you mean no more 10m every year type sales....

I'd say after 4 more iterations, sales will decline below the 10m range...so 2015.

If they keep bringing one out every year then probably about 4-5 years time. It'll still sell, just not as ridiculously as it does now. If Activision decide to release more than 1 CoD a year then shorter.

The FPS genre was around and popular way before Call of Duty. What else will people fall back to? Halo, Half-Life, Counterstrike, Left 4 Dead, Day of Defeat, Deus Ex, Unreal Tournament, Quake, Doom, Rage, Duke Nukem, Bioshock, Bulletstorm etc.

And videogames would be fine too.

This year or next year, it'll start declining, prolly, unless it's immortal, which it might be. It'll burn out when the HD sequels release since I don't think it'll adapt so 2012/13, I expect.