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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, for example).