You have to say what is the meaning of $\Delta u$, and of $\int\Delta u$. For the integral to have a meaning, $\Delta u$ must be a measure. Then $\int_M\Delta u=0$
without any assumptions about $C^2$ or $S$. (This is called "Main Theorem"
in S. Donaldson, Riemann surfaces, Chap 8, Thm. 5).

But if you understand $\Delta u$ in the naive, classical sense for $C^2$ function, and understand your integral as $\int_{M\backslash S}\Delta u$ then of course this does not have to be $0$.

For example, on the Riemann sphere, let $u(z)=0,|z|<1\; u(z)=|z|, 1<z<2\; u(z)=2, |z|>1$. This function is $C^2$ except on the set of measure zero
consisting of two circles $|z|=1$ and $|z|=2$. But
$$\left(\int_{|z|<1}+\int_{1<|z|<2}+\int_{|z|>1}\right)u(z)>0.$$