A couple of years ago, David Speyer showed me the following counterexample.

Let $E$ be an elliptic curve defined over a field, with points $P$, $Q$ that are linearly independent under the group law. Let $p:X\to E$ be the total space of the rank two vector bundle $\mathcal{O}([P])\oplus\mathcal{O}(-[Q])$ on $E$. Then $X$ is is finite type, but it can be checked that the ring $\Gamma(\mathcal{O}_X)$ of global regular functions on $X$ is not finitely generated.

If we pick an affine open cover $\{U,V\}$ of $E$, then $\{p^{-1}(U),p^{-1}(V)\}$ is an affine open cover of $X$. The rings of regular functions on $p^{-1}(U)$ and $p^{-1}(V)$ are finitely generated, but their intersection inside the function field of $X$ (or inside the finitely generated subalgebra they generate) is $\Gamma(\mathcal{O}_X)$, which is not finitely generated.

A couple of years ago, David Speyer showed me the following counterexample.

Let $E$ be an elliptic curve defined over a field, with points $P$, $Q$ that are linearly independent under the group law. Let $p:X\to E$ be the total space of the rank two vector bundle $\mathcal{O}([P])\oplus\mathcal{O}(-[Q])$ on $E$. Then $X$ is finite type, but it can be checked that the ring $\Gamma(\mathcal{O}_X)$ of global regular functions on $X$ is not finitely generated.

If we pick an affine open cover $\{U,V\}$ of $E$, then $\{p^{-1}(U),p^{-1}(V)\}$ is an affine open cover of $X$. The rings of regular functions on $p^{-1}(U)$ and $p^{-1}(V)$ are finitely generated, but their intersection inside the function field of $X$ (or inside the finitely generated subalgebra they generate) is $\Gamma(\mathcal{O}_X)$, which is not finitely generated.