Let $X=X_\Sigma$ and $X'=X_{\Sigma'}$ be toric varieties such that the fan $\Sigma'$ is a refinement of the fan $\Sigma$, and denote by $f:X'\to X$ the corresponding toric morphism.
Let $D=\Sigma a_\rho D_\rho$ be a torus invariant divisor on $X$, where the sum is indexed by the rays of the fan $\Sigma$, so $D_\rho$ is the closure of the codimension 1 orbit corresponding to $\rho$ under the cone-orbit correspondence. Now since $\Sigma'$ is a refinement of $\Sigma$, every ray in $\Sigma$ is a ray in $\Sigma'$ (right? otherwise assume this happens), so we have a torus invariant divisor $D'=\Sigma a_\rho D'_\rho$ on $X'$, where the sum is indexed by the rays of the fan $\Sigma$.
Now consider the coherent sheaf $L:=\mathcal{O}_X(D)$ and its pullback $L':=f^*L$. Is $L'\cong\mathcal{O}_{X'}(D')$?
My real question is how to find a divisor corresponding to $L'$, but I formulated the question like this because $D'$ is my naive guess. Feel free to assume all divisors are Cartier, so the sheaves are invertible.
For the record: the answer to my question is no. The correct way to pullback Cartier divisors in this context can be extracted from the proof of Proposition 6.2.7 of the book Toric Varieties by Cox-Little-Schenck.