Is it true that $$\mathrm{H}^p_\mathrm{Zar}(\mathbb{A}^n_\mathbb{K}, \mathcal{F})=0$$ for any $p>1$ and $\mathcal{F}$ (non constant) sheaf of abelian groups?
If not, is it true for some fields/sheaves but not others? Which ones?
Do you know some counterexamples?
I'd really appreciate some reference, thank you in advance.
No, this is not true unless you restrict to quasicoherent sheaves. For instance, just as a topological space, $\mathbb{A}^1$ is homeomorphic to $\mathbb{P}^1$, and there are coherent sheaves on $\mathbb{P}^1$ with nontrivial $H^1$. In fact, there are even sheaves of $\mathcal{O}_{\mathbb{A}^n}$-modules on $\mathbb{A}^n$ which have nontrivial cohomology. For instance, let $p\in\mathbb{A}^n$ be a closed point and let $\mathcal{G}$ be the skyscraper sheaf at $p$ on $\mathbb{A}^n$ with stalk $\mathcal{O}_{\mathbb{A}^n, p}$. This sheaf has a global section corresponding to $1\in\mathcal{O}_{\mathbb{A}^n,p}$ and there is a corresponding map of sheaves $\mathcal{O}_{\mathbb{A}^n}\to\mathcal{G}$. In fact, this map is surjective, as is easily seen by looking at stalks. If you let $\mathcal{F}$ be its kernel, then there is an exact sequence $$H^0(\mathcal{O}_{\mathbb{A}^n})\to H^0(\mathcal{G})\to H^1(\mathcal{F}).$$
But the map $\mathcal{O}_{\mathbb{A}^n}\to\mathcal{G}$ is not surjective on global sections, so it follows that $H^1(\mathcal{F})\neq 0$.