Suppose $B$ is path-connected, and admits the structure of a CW-complex. If there are two fibrations $F_1\rightarrow E_1\rightarrow B$ and $F_2\rightarrow E_2\rightarrow B$ such that $F_1$ and $F_2$ are homotopy equivalent, I was wondering whether if it was true that $E_1$ and $E_2$ are homotopy equivalent, if not what conditions would be required?
P.S. This is not a homework question but my curiosity.
It is quite false that this implies homotopy equivalence. For a first example, take the $\mathbb{Z}/2\mathbb{Z}$-bundles over a circle given by (a) two disjoint circles and (b) stacking two circles on top of each other, cutting them above one point, then swapping strands and gluing. Both are $\mathbb{Z}/2\mathbb{Z}$ bundles over the circle, but only one is connected.
There are a great many invariants that can be used to distinguish fiber bundles in various settings with the same base and same fibers, and at times classify them. A good phrase to look up would be "characteristic classes," though the literature on these things can be rather dense.