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](http://dx.doi.org/10.5169/seals-87871)). 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 $\overline{\phi}:(A\ast B)/\operatorname{ncl}(w)\twoheadrightarrow(A\ast\mathbb{Z})/\operatorname{ncl}(u)$ for some word $u\in A\ast\mathbb{Z}$. 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.
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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](https://mathoverflow.net/q/2372/6503). The paper of Fenn and Rourke I cited above is the best place to read about it though.
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Added later: I asked Jim Howie if he knew what the status of this problem is, and he said he thought it open and very difficult (so there are no *known* counter-examples).
He also pointed to two papers of his - one with Brodskii from 1993, and the other with Edjvet from 2021. These prove that when $A,B$ are torsion-free, then they embed into $(A*B)/\operatorname{ncl}(w)$ if $w$ has free product length $\le 6$ (first paper) and $\le 8$ (second paper). These are innocuous-sounding results, but the one extra step took ~28 years, and both took quite a bit of work to prove.
Embedding results like these are typically referred to as *Freiheitssatz* and seem to be the main approach to problems of this form. For example the main result of Klyachko's paper is a Freiheitssatz for when $w$ has a specific form; the other cases follow from an easy observation. A Freiheitssatz is a priori stronger than a non-triviality result would be, however Jim points out that it is not clear how much stronger.
The papers are:
S. D. Brodskii and J. Howie, *One-relator products of torsion-free groups*, Glasgow Math. J. 35 (1993) 99-104 ([doi](https://doi.org/10.1017/S0017089500009617)).
M. Edjvet and J. Howie, *On singular equations over torsion-free groups* Internat. J. Algebra Comput. 31 (2021), 551-580 ([doi](https://doi.org/10.1142/S0218196721500272), [arXiv](https://arxiv.org/abs/2001.07634.pdf).