I am gonna use my PC on a 24" screen and also on another 16" screen for other stuff. I almost always use my PC for gaming on games such as BF3 and Skyrim.
The performance may suffer, if you play on both monitors at the same time. Otherwise, if you just play on one screen, and just browse on the other it will be fine.
No matter how big or small the screen is, the only thing that would affect the performance of a graphics card is the resolution you will be playing at, the graphic settings and if you do multiple monitors. If you play at 1920x1080 or higher the performance will go down.
The performance may suffer, if you play on both monitors at the same time. Otherwise, if you just play on one screen, and just browse on the other it will be fine.