The answer to the question as asked is no: a fibre-preserving map of fibrations in which the maps of total and base spaces are homotopy equivalences is neccessarily a fibre-preserving homotopy equivalence (also known as a homotopy equivalence of fibrations). A reference was supplied in the comments by Gustavo Granja, to Peter May's book A Concise Course in Algebraic Topology, where the statement appears as a Proposition on page 53. (The proof, although not given in detail, does appear to be a straightforward dualization of the corresponding result for cofibrations, proved on page 48.)