During a lecture my professor made an observation, she said that it can happen the following fact but she never explained how/why: let $\mathcal{A}$ a category; there are examples of morphisms $f\colon A^\bullet\to B^\bullet$ of complexes such that the morphism induced by $f$ in the derived category $D(\mathcal{A})$ is zero but the morphism induced by $f$ in the homotopy category $K(\mathcal{A})$ is not zero.
I'd like to understand how this can happen. Can anyone give me any examples please?
I'd like to precise that I know things like cone, cylinders, exact triangles.