I've tried that, and they tend to just lock up the position with f4.
King's Indian Defence - how to push f5?
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I'm no KID expert, but white is trading off his e pawn, so at a very basic sense my guess would be is that it surrenders some of the center and black gets better piece play.
Wait... As I type this, I remembered that there is a rule I remember hearing: "take with the g pawn in the KID" I suppose if you do that, you get the same attack with a free open g-file.
exf5 is not bad exactly, and this is not black and white: exf5 does have it's strong points, but ultimately in opening theory it doesn't give white good chances for the advantage so it's not played. The idea of exf5 is, if ...gxf5 isn't played, to clear e4, and this blockade does make it tough for black to attack with only an open f file.. If ...gxf5 is played (the correct move, keeping control of the center and opening the g file for attack) he often tries to close things up and attack the structure with something like f4 (trying to prove the center pawns to be a drawback instead of a strength). This is double edged, because black can at some point play ...e4, weakening d4, but gaining space AND opening up the g7 bishop. Or he can open things with ...exf4.
I don't know the exact analysis after exf5, I just know that according to opening theory it leads to approximate equality. White's f4 ideas are just not quite enough.
Your intuition is right. There's a reason that pawn goes unmolested so often in master play.
Many lines of the KID feature black with a pawn chain running c7-d6-e5. Those are usually locked up with white pawns on d5 and e4.
That closed position benefits white in two ways: 1) it gives white a lasting queenside space advantage; 2) it effectively entombs the dark square bishop on g7. Now, that bishop is doing valuable service where it is, in helping control f4, of course. But its tactical and strategic potential goes through the roof if the black pawn on e5 moves or disappears.
In a game that features that pawn structure, capturing exf5 allows black to advance the e pawn, freeing the bishop and controlling the long diagonal that runs through the heart of what moments ago was white's sovereign territory on the queenside. The plusses of the structure turn to minuses in the flick of a pawn.
Doesn't mean white's game is lost, but it throws an awfully big monkey wrench into his strategic plans.
exf5 is not as rare as you seem to think. I've seen many GM level games where white takes there but of course GMs understand exactly under which circumstances such a trade is a viable option. If black then has a choice of recapturing with a pawn or a piece it's indeed more common to capture with the g-pawn but there are exceptions... One of the main reasons why black often rather captures with a pawn is to control e4-square which is often a key square in KID. Incidentaly one of the possible reasons why white in some positions might want to play exf5 is that he subsequently hopes to make use of e4 by planting a knight there.
Note that in the classical variation it's typical to recapture f5 with a knight.
Yeah ...Nxf5 is often a good option, that is if it's available.
Your damn right it doesn't 
Well actually on some occasions ...f5 can be played just to liquidate the center (long after the e pawn exchanges for the d pawn), but because of the weaknesses it creates and the lines it opens for white often it isn't good.
Excellent explanation by Shakaali.
Ok, I love to play the King's Indian, but whenever I push f5 to begin my attack white just takes (admittedly, activating my bishop with Bxf5), but this seems to more or less kill my attack. In most high level games, white does not take this pawn. What is the disadvantage to white in taking this pawn?
I know I don't have a specific KID variation specified, feel free to use any variation in your example.
EDIT: I'm also wondering what to do against the four pawn attack by white (where white pushes f4 as well)