You might want to look up *quasifibrations*, which are surjective maps $p\colon E\to B$ such that $p\colon (E,p^{-1}(b))\to (B,b)$ is a weak equivalence for all $b\in B$. Any quasifibration gives rise to a long exact sequence as in your question. There are certainly examples of quasifibrations which aren't fibrations (see Mike Shulman's comment to [this question][1], for example).


  [1]: http://mathoverflow.net/questions/60763/when-a-quasifibration-is-a-hurewicz-fibration