From $0 = f(f(0)) = f(b) = ab + b = (a+1)b$, we get $a = -1$ or $b = 0$.
If $b = 0$, i.e. $f(x) = ax$, then we have $9 = f(f(f(4))) = f(f(4a)) = f(4a^2) = 4a^3$, which has no solution as $a$ is an integer.

So we must have $a = -1$. Then $f(x) = -x + b$, and hence $f(f(x)) = -(-x + b) + b = x$. It follows that $f(f(f(x))) = f(f(x)) = x$ for all $x$, and thus the answer is
$$1 + 2 + \cdots + 2014 = \frac{2014 \times 2015}{2} = 2029105.$$