Here is a partial answer: The Kervaire-Laudenbach Conjecture states that, for any group $A$, $(A\ast\mathbb{Z})/\operatorname{ncl}(w)$ is non-trivial. This was proven by Klyachko for torsion free groups $A$; see Fenn and Rourke, Klyachko's methods and the solution of equations over torsion-free groups, Enseign. Math. (2) 42 (1996), no. 1-2, 49–74 (doi). Hence:
There are no counter-examples for $A$ torsion-free and $B\cong Z$.
This can be easily generalised: A group is indicable if it surjects onto $\mathbb{Z}$. If $B$ is indicable, so $\phi:B\twoheadrightarrow\mathbb{Z}$, then the map $\phi$ induces a map $\overline{\phi}:(A*B)/\operatorname{ncl}(w)\twoheadrightarrow(A*\mathbb{Z})/\operatorname{ncl}(u)$ for some word $u\in A*\mathbb{Z}$. By Klyachko's theorem, we know the image is non-trivial so we immediately have:
There are no counter-examples for $A$ torsion-free and $B$ indicable.
Klyachko's proof is really pretty, based on a "funny property" of spheres which "is so simple and funny" that, he claims, it could be included in a collection of puzzles or suggested as a problem for a school mathematical tournament. He phrased it in terms of car crashes, and you can find a MathOverflow discussion on it here. The paper of Fenn and Rourke I cited above is the best place to read about it though.