Let $S$ be a smooth projective algebraic surface over $\mathbb C$ and $C$ be a smooth curve on $S$. Is it always true that $dim_{\mathbb C} H^0(S,O_{S}(C))=0$ ? In particular, is it zero when $S$ is a K3surface ?

I'm not sure about the general case, but this is at least true if the divisor associated to the curve C is ample: In this case the line bundle L associated to the divisor is ample, so the KodairaNakano vanishing theorem applies. By Serre duality we get $$ 0 = H^{2,1}(S,L) = H^1(S, \Omega^2 \otimes L) = H^1(S,L*)^* = H^1(S,O(C))^* $$ as the dual of $L$ is the line bundle associated to the divisor $C$. Remember that $\mathcal I_C = O(C)$, then the long exact sequence associated to $$ 0 \to O_S(C) \to O_S \to O_C \to O $$ gives the exact sequence $$ 0 \to H^0(S,O_S(C)) \to \mathbb C \to \mathbb C \to 0 $$ as both $O_S$ and $O_C$ only have constant global sections. The vanishing of $H^0$ follows. The general case would hold in the same way if $H^1(S,O(C)) = H^1(S,\mathcal I_C) = 0$ for any smooth curve $C$ in $S$. 

