Is there an analogue of Ehresmann fibration theorem for (finite) CW complexes ?

Note is not true that an open surjective (necessary proper) cellular map of finite CW or simplicial complexes is necessarily a Serre fibration: ramified coverings of surfaces are counterexamples, as pointed out in comments.

Relevant questions are: Ehresmann fibration theorem for manifolds with boundary, and Ehresmann's fibration theorem in the C1 class, and Simplicial approximation of a fibration, and Which maps of simplicial sets geometrically realize to fibrations?.