I don't know the answer in general, but in the case that we (a) restrict to compactly-generated weak Hausdorff spaces (not a big deal), and (b) restrict to the case where F is compact, I think I have an answer.
Namely, by a result of May there is a space, $BG$, with the homotopy type of a CW-complex (since $F$ has this property), such that fibrations over $B$ with fiber having the homotopy type of $F$ are classified up to equivalence by homotopy-classes of maps $B \rightarrow BG$.
More specifically, there is a universal fibration $EG \rightarrow BG$ with fiber $F$, and pulling back this fibration by the map $B \rightarrow BG$ gives the fibration you started with (up to equivalence). Since $EG$ also has the homotopy type of a CW-complex, then the total space will as well.
So if a counterexample exists, $F$ will have to be non-compact.