The solutions to the Dirichlet problem of elliptic PDE with smooth enough coefficients below to H_0^1 and also belong to C_infinity on the interior. Does that mean the classical limit of the function on the boundary is 0?

Yes. By Theorem 9.17 in Brezis' book, let $\Omega$ be just a bit regular$^*$ and let $u\in W^{1,p}(\Omega)\cap C(\bar{\Omega})$. Then $u\in W^{1,p}_0(\Omega)$ if and only if $u(z)=0$ for all $z\in \partial \Omega$. $^*$ $C^1$ will do, but even less regularity might be really necessary, I guess. 

