We certainly don't want to plug the function into itself $2012$ times, so we test the first few to see if there is a pattern:
$f^1(x) = f(x) = \frac{x-\sqrt3}{x\sqrt3+1}$
$f^2(x) = \frac{\left(\frac{x-\sqrt3}{x\sqrt3+1}\right)-\sqrt3}{\left(\frac{x-\sqrt3}{x\sqrt3+1}\right)\sqrt3+1} = \frac{x+\sqrt3}{1-x\sqrt3}$
$f^3(x) = \frac{\left(\frac{x-\sqrt3}{x\sqrt3+1}\right)+\sqrt3}{1-\left(\frac{x-\sqrt3}{x\sqrt3+1}\right)\sqrt3} = x $
This means that $f^4(x) = f(x)$, so the terms repeat every $3$.
$2012/3=670 \text{ remainder } 2$, so $f^{2012}=f^2 = \boxed{\frac{x-\sqrt3}{1-x\sqrt3}}$.