Let $X$ be a smooth projective surface over $\mathbb{C}$ and $E$ be a vector bundle on it. Let $F\subset E$ be a normal subvector bundle of $E$, that is, the quotient $E/F$ is torsion free. Any section $s\in H^0(F)$ naturally defines a section of $E$, that I call $s\_E$. Does the zero locus of $s\_E$ always coincide with that of $s$?



Sections of a normal subsheaf.
If you mean that you have a morphism of sheaves $\mathcal O\_X(F)\to\mathcal O\_X(E)$ such that the quotient sheaf is torsion free, then in general the answer is no. 


Think at the example of a holomorphic foliation $\mathcal F$ by curves on a smooth projective surface $X$. It is given by definition by the datum of a holomorphic line bundle $T\_\mathcal F\to X$ together with a sheaf morphism $\varphi\colon\mathcal O\_X(T\_\mathcal F)\to\mathcal O\_X(T\_X)$, such that the quotient is torsion-free. Thus, you have a short exact sequence of sheaves 
$$
0\to\mathcal O\_X(T\_\mathcal F)\to\mathcal O\_X(T\_X)\to\mathcal I\_Z\cdot\mathcal O\_X(N\_\mathcal F)\to 0,
$$
where $N\_\mathcal F$ is a holomorphic line bundle and $\mathcal I\_Z$ is an ideal sheaf supported on a zero dimensional cycle. This ideal sheaf detects exactly the set of point where the morphism you started with is not injective (and at these point it is in fact the zero morphism). 


Thus a global holomorphic section $s\in H^0(X,T\_\mathcal F)$ will vanish at those points, once sent in $H^0(X,T\_X)$ by the morphism $\varphi$.


On the other hand, if you really meant subbundle, then the morphism is just the inclusion, the quotient is actually a vector bundle (hence torsion-free) and of course the zero loci always coincide. 


