Fundamental group of a complex algebraic curve residually finite? Is the analytic fundamental group of a smooth complex algebraic curve (considered as a Riemann surface) residually finite?

Yes. Recall that topologically such a surface is a $g$-holed torus minus $n$ points. Except in the cases $(g, n) = (1, 0), (0, 0), (0, 1), (0, 2)$ such a surface, call it $S$, has negative Euler characteristic, so by the uniformization theorem its universal cover is the upper half plane $\mathbb{H}$. Since the action of $\pi_1(S)$ on $\mathbb{H}$ by covering transformations is an action by biholomorphic maps, $\pi_1(S)$ embeds into $\text{PSL}_2(\mathbb{R})$. And any finitely generated subgroup of $\text{PSL}_2(\mathbb{R})$ is residually finite; the argument is nearly identical to the argument that any finitely generated linear group is residually finite. 
The exceptional cases are straightforward to verify individually.
