The S value is more of an issue than the mAh rating. If you use, for example, a 4S battery in a plane that's rated for 3S, the additional voltage may overload the ESC or motor. A larger mAh rating, though, simply will give a longer flight time than a smaller rating. The ESC/motor combo will pull the same amps at a given throttle setting, regardless of the mAh rating of the battery. The other factor to consider is the C rating. A higher C rating means the battery can deliver a higher output current, which can be important if you're flying a model with a high performance propulsion system.
I've flown many of my C pack models on 3S 2200 mAh batteries, and have never had an issue with them.