Supose $X$ is a compact Hausdorff topological space and $Y \subseteq X$ a dense subset. If $S$ is a set of functions in $C(X)$ such that separates points in $Y$ then the Stone-Weierstrass theorem still hold?
No, this never works unless $Y$ is all of $X$. Indeed, suppose $x\in X\setminus Y$ and $y\in Y$. Consider the subalgebra $A$ of $C(X)$ consisting of all functions $f$ such that $f(x)=f(y)$. Then $A$ is a closed *-subalgebra of $C(X)$ and it separates points of $Y$, but it is not all of $C(X)$.