I came across the following result.
Let $X$ be a separated $k$-scheme that admits an open covering of spectra of finitely generated $k$-algebras. Let $f:Y\rightarrow X$ be a morphism of schemes which is affine and of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_{Y}$-module. Then we have natural isomorphisms of $k$-vectorspaces $$H^{i}(Y,\mathcal{F})\cong H^{i}(X,f_{*}\mathcal{F}).$$
I would like to see an example of a morphism $f:Y\rightarrow X$ of separated $k$-schemes and a quasi-coherent $\mathcal{O}_{Y}$-module $\mathcal{F}$ for which there exists an integer $i$ such that $$H^{i}(Y,\mathcal{F})\ncong H^{i}(X,f_{*}\mathcal{F}).$$
To close this post I repeat here the counter example Mindlack and Roland gave in the comments.
Take $X = \operatorname{Spec}(k)$ and $Y=\mathbb{P}_{k}^{1}$ and $\mathcal{F}=\mathcal{O}_{Y}(-2)$. Then one can find for $i=1$ that we indeed do not have an isomorphism.