Let $x \in X$ be a (edit: closed) point in a scheme. If $X$ is sufficiently nice (e.g noetherian and integral or even a variety), can we write $H^0(X \backslash \{x\}, \mathcal{O}_{X \backslash \{x\}})$ as some kind of localization/ in terms of $H^0(X,\mathcal{O}_X)$? Apologies if this is obvious, I'm a bit rusty here.
In the affine case if $X = \text{Spec} \ R$, I can see $X \backslash \{x\} = D(f_1) \cup \dots \cup D(f_l)$ for some elements $f_i$.
Proposition 3.29 in the book Algebraic Geometry by Görtz/Wedhorn states that for an integral scheme $X$ and an open subset $U$, we have:
$$\mathcal O_X(U) = \bigcap_{z \in U} \mathcal O_{X,z}$$ , where the intersection is taken in the function field of $X$.
In your particular case, this tells us
$$\mathcal O_X(X \setminus \{x\}) = \bigcap_{z \neq x} \mathcal O_{X,z}.$$
If $X$ is at least two-dimensional, this intersection will run through all local rings at co-dimension one points. In particular if $X$ is also normal, the intersection equals $\mathcal O_X(X)$ by Hartog's lemma.