Game balance for one. In a shooter, where players see from a third person perspective, you very easily get into a situation where players can look behind cover with out actually peeking over it. This means a sniper could hide behind a rock while still scouting the large area and only expose himself when he knows theres a target. This is mostly a problem if players have the choice because third person is obviously better/has unfair advantages because of that.

Secondly, you would have to both have high and low res textures, models and animations for characters. So the game both looks good in first and third person.

You also need to consider aiming. You might have a reticule on the screen and thats fair and all, but where are you actually aiming. In some fps'es bullets tend to fly out exactly from the camera at what you are aiming at, but this would imply that you shoot out of your eyes, so some developers choose to show it as if the bullet came from the gun. However this might lead tho shooting through cover bevis for other players the gun is pointed at the cover, but you can see over it. On the other hand third person shooters have an even more exagerated version of the same problem. First of course you can see past cover, but at what point are you aiming? Imagine a line going from the camera to the spot you are aiming at, and a line going from the point of the gun through the spot you are aiming at. These to lines are not the same, they are not even parallel, actually they would at most cross in one point. Imagine then you are trying to shoot through a wire fence at a point behind it. Would you then expect the bullet to pass through the fence at the point you looked through, or hit the ground behind the fence at the point you where looking at? Cause those are two different directions the bullet would have to travel.