The answer is no.
Any surface of degree $d$ passing through the lines is of the form $l_1 Q_1+l_2Q_2$ where $Q_1,Q_2$ are homogeneous polynomials of degree $d-1$.
Since the line does not intersect the curve $C$, the curve $C$ is not in a plane passing through the line. Hence, neither $l_1$ neither $l_2$ vanishes on $C$. But this does not imply that $Q_1,Q_2$ vanish on $C$.
Edit: I gave an explicit counterample before, but there was a mistake in it, as observed by Mohan.