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).

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).