Answer
The function is not differentiable on the interval $(0,\pi)$. The hypotheses of Rolle's Theorem are not satisfied.
Work Step by Step
$f(x) = tan ~x$
$f(0) = tan~0 = 0$
$f(\pi) = tan~(\pi) = 0$
Thus $f(0) = f(\pi)$
$f'(x) = \frac{1}{cos^2~x}$
The derivative does not exist when $x = \frac{\pi}{2}$. Thus, the function is not differentiable on the interval $(0,\pi)$. The hypotheses of Rolle's Theorem are not satisfied.
Therefore, the fact that there is no number $c$ in the interval $(0,\pi)$ such that $f'(c) = 0$ does not contradict Rolle's Theorem.