Let $A$ be a finitely generated $k$-algebra, where $k$ is a field, let $I$ be an ideal in $A$, let $M$ be a finitely generated $A/I$-module, and let $M^{\prime}$ denote $M$ considered as an $A$-module. Let $B$ be a finitely generated $A$-algebra. Is it true (perhaps under some additional conditions on $A$ and $I$, though I need the case when $B$ is not flat over $A$) that $M^{\prime} \otimes^L\_A B$ and $M \otimes^L\_{A/I} B/IB$ are quasi-isomorphic as complexes of $B$-modules? 



Base change for non-flat coherent sheaves and affine maps
There is an associativity identity for total derived tensor products, cf. [Stacks Project Tag 08YU](http://stacks.math.columbia.edu/tag/08YU). In your case, for the triple of rings, $$A\twoheadrightarrow A/I \xrightarrow{\text{Id}}A/I,$$ this gives an equivalence in the derived category, $$ M\otimes\_{A/I}^{\textbf{L}}(A/I\otimes\_A^{\textbf{L}} B) \cong M\otimes\_A^{\textbf{L}} B. $$ Thus, if the natural truncation morphism, $$A/I\otimes\_A^{\textbf{L}}B \to h\_0(A/I\otimes\_A^{\textbf{L}}B), \text{ i.e., }\ A/I\otimes\_A^{\textbf{L}}B \to B/IB, $$
is an equivalence in the derived category, then that gives the equivalence that you are asking about, $$M\otimes\_{A/I}^{\textbf{L}} B/IB \cong M\otimes\_A^{\textbf{L}}B.$$


As the OP points out, the natural truncation morphism is an equivalence in the derived category if and only if $h\_i(A/I\otimes\_A^{\textbf{L}} B)$ vanishes for every $i>0$, i.e., if and only if $\text{Tor}\_i^A(A/I,B)$ vanishes for every $i>0$. The OP asks whether it might suffice to have vanishing of $\text{Tor}\_i^A(A/I,M)$ for $i>0$? It is hard for me to imagine any situation where this vanishing would hold, since $\text{Tor}\_1^A(A/I,M)$ is naturally isomorphic to $I\otimes\_A M$ for every $A/I$-module $M$. 


