All the current answers are sort of alluding to the following, but not saying it explicitly.

In any setup where you have the six functors (de Rham, Betti = sheaves of vector spaces on topological spaces, $\ell$-adic cohomology,... ) and $i:Z\to X$ is a closed embedding, then the local cohomology is defined as $$\mathcal{H}_Z(X,\mathcal{F})\ =\ \text{H}^*(Z,i^!\mathcal{F})$$ so we have an analogue of $\text{H}^*(X,X\setminus Z)$ for any of your favourite cohomology theories (or others like stable homotopy theory, algebraic cobordism, K theory, Chow, ... ). Then using all the properties of the six functors (Gysin sequence etc), you get the standard e.g. long exact sequences involving local/relative cohomology.

But a warning: this does not include $\text{QCoh}$, which does not have a six functor formalism I think.

All the current answers are sort of alluding to the following, but not saying it explicitly.

In any setup where you have the six functors (de Rham, Betti = sheaves of vector spaces on topological spaces, $\ell$-adic cohomology,... ) and $i:Z\to X$ is a closed embedding, then the local cohomology is defined as $$\mathcal{H}_Z(X,\mathcal{F})\ =\ \text{H}^*(Z,i^!\mathcal{F})$$ so we have an analogue of $\text{H}^*(X,X\setminus Z)$ for any of your favourite cohomology theories (or others like stable homotopy theory, algebraic cobordism, K theory, Chow, ... ). Then using all the properties of the six functors (Gysin sequence etc), you get the standard e.g. long exact sequences involving local/relative cohomology.