Let $C$ be a non-contractible complex. Let $X$ be a direct sum of a countably infinite number of copies of $C$ plus a countable infinite number of copies of $\Sigma C$. Then the inclusion of $C$ into $X$ as a direct summand, and the null map from $C$ to $X$, are non-isomorphic maps with isomorphic mapping cones. I think.

Even if it works, this example feels like a swindle. Is there one with finitely generated modules?

Let $C$ be a non-contractible complex. Let $X$ be a direct sum of a countably infinite number of copies of $C$ plus a countable infinite number of copies of $\Sigma C$. Then the inclusion of $C$ into $X$ as a direct summand, and the null map from $C$ to $X$, are non-isomorphic maps with isomorphic mapping cones.

Even if it works, this example feels like a swindle. Is there one with finitely generated modules?