What is the higher categorical generalization of exact sequence (3 terms or $\mathbb{Z}$ terms)? In particularly, consider the simplest cases: chain complexes, $L_\infty$algebras.

I'm not sure exactly what you're asking for, but the notion of distinguished triangle in a triangulated category (or, if you want to be honestly highercategorical, a stable $(\infty,1)$category) is pretty close. Exact sequences as such only make sense in an abelian 1categorical context, and the highercategorical analogue of an abelian category is a stable $(\infty,1)$category. In a distinguished triangle, the final term is the "cokernel" of the first two and the first term the "kernel" of the second two, and there is an induced long exact sequence on homology/homotopy. (The distinction between short exact and long exact sequences blurs in higher category theory, since any triangle can be extended indefinitely and doesn't zero out  the kernel of a kernel isn't usually zero, and so on.) There are other notions of "exactness" in nonabelian settings. 

