Let $X$ be a closed, orientable surface of genus at least 2, and let $\phi: \pi_1(X) \to \pi_1(X)$ be a surjective homomorphism. Is $\phi$ necessarily injective?

Yes. Surface groups are Hopfian. More generally, all residually finite groups are Hopfian  see Theorem IV.4.10 in Lyndon and Schupp's book "Combinatorial Group Theory". 

