Does anything happen if I forget and tensor back up along a highly connective map of $E_{\infty}$-rings?

Here's what I mean precisely: Let $f \colon A \to B$ be a $n$-connective map between connective $E_{\infty}$-rings, with $n \geq 1$. Here $n$-connective means that $\pi_i (fib(f))=0$ for $i < n$. Roughly, $f$ is a very surjective map, and if we let $fib(f)=I$, then $B$ is roughly $A/I$.

Now let's take a $B$-module $M$. We can forget the $B$-module structure and view it as an $A$-module. Then we can tensor it back up again to get $B \otimes _A M$. By adjunction there is a canonical map $m \colon B \otimes _A M \to M$, which is the multiplication map. How far is this map from being an equivalence?

The case I am really interested is when $f \colon A \to B$ is a square-zero extension obtained from a $n$-connective derivation $\eta \colon L_B \to M[1]$. Then $fib(f)$ can be identified with $M$, and so additonally has the structure of a $B$-module. Does anything special happen in this case?