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).
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. 

