Hope this helps, it doesn't give a definite answer, but it tells about where to find an obstruction.
First off, the existence of multiple fibers in an elliptic fibration is an obstruction to the existence of a differentiable section (over $\mathbb{C}$). On the other hand we can always get rid of the multiple fibers by passing to an \'etale cover.
Now a more elaborate answer, given an elliptic fibration with no sections $X\rightarrow B$ we can associate a fibration $J\rightarrow B$ which has a section and a rational map $\phi:J\times_BX\rightarrow B$ that commutes with projections to $B$ and has certain properties. The family $J$ is called jacobian family.
The elliptic fibrations are classified by their jacobian fibrations and here comes the change of quantifiers. One can introduce a group structure on the set $I(J)$ of elliptic fibrations with a given jacobian fibration. Hence if the class of the fibration $X\rightarrow B$ in $I(J)$ is not zero, (the obstruction) the fibration $X$ has no differentiable sections. This works the same way the the first Chern class of a line bundle $L$ gives and obstruction for $L$ to be trivial.