First, note that what you want isn't true even in dimension 2, unless you specify that $B$ is minimal. Otherwise one can take a $K3$ surface say and blow up a point: the resulting surface certainly still has $\kappa=0$, but its canonical bundle is $E$, the exceptional divisor, which isn't torsion.
So we should restrict to the case where $B$ is minimal, in other words, that $K_B$ is nef. Then the abundance conjecture (which is a theorem in dimension $\leq 3$) says that $K_B$ is semi-ample, in other words some multiple $mK_B$ is basepoint-free and gives a morphism to a variety of dimension $\kappa(B)$. In this case that implies that $mK_B$ is pulled back from some line bundle on a point, hence is trivial, so $K_B$ is indeed torsion.
On the other hand, if we knew your statement for all minimal $B$ of Kodaira dimension 0, then we would know the abundance conjecture in that case.
But that is still very much open.
Update: I just realised the last sentence is completely false. Kawamata proved the abundance conjecture for minimal varieties with $\kappa=0$ in 1985!