Is a fibre bundle with contractible fibre a homotopy equivalence? As already stated in the title: If $p:E\to X$ is a fibre bundle with a contractible fibre $F$, is $p$ a homotopy equivalence? Or are there stronger conditions we need, e. g. the existence of a section $s:X\to E$?

If the spaces involved are CW-complexes*, then yes. As $F$ is contractible, the long exact sequence in homotopy groups shows that $p_*: \pi_n(E) \to \pi_n(X)$ is always an isomorphism. The Whitehead theorem (that a weak h.e. of CW-complexes is an equivalence) now shows that $p$ is a homotopy equivalence.

*or even homotopy equivalent to CW-complexes