Let $X$ be a smooth projective variety of dimension $\geq 2$ and $E$ a vector bundle on $X$ of rank $r\geq 2$. Is it true that, if $E$ is globally generated, then the zero locus of a general section of $E$ has codimension $\geq r$ in $X$?

As Donu pointed out, it may happen that the general section has no zeroes. Anyway, the the answer to your question is yes. This is a special case of the following more general result "of Bertini type" about degeneracy loci of morphism of vector bundles, whose proof can be found in Ottaviani's book "Varietà proiettive di codimensione piccola" (in Italian, but I guess that many references in English are also available). THEOREM. Let $\phi \colon F \to E$ be a morphism between vector bundles on $X$, with $\textrm{rank} \, F=s$, $\textrm{rank} \, E=r$, and let $D_k(\phi)$ be the set of points $x \in X$ such that $\phi_x \colon F_x \to E_x$ has rank $\leq k$. If $F^{*} \otimes E$ is generated by global sections, then for the general $\phi$ either $D_k(\phi)$ is empty or it has the expected codimension $(sk)(rk)$. Moreover, if $F^{*} \otimes E$ is ample and $(sk)(rk) < \dim X$, then for the general $\phi$ the locus $D_k(\phi)$ is nonempty, of the expected codimension $(sk)(rk)$. Now apply this result to the morphism $\phi \colon \mathcal{O}_X \to E$ induced by the section. Since $D_0(\phi)$ is exactly the locus where the section vanishes, you are done. 

