Is the global section ring of a Noetherian Scheme Noetherian as well?
The answer is No. See the note Un ouvert bizarre by Manuel Ojanguren.
Here is an outline of the construction: Let $A,B \subseteq \mathbb{P}^3_k$ be two projective planes which intersect in a projective line $L$. Let $X = A \cup B$. Let $D \neq L$ be a projective line on $A$ with $D \cap L = \{P\}$. Let $U = X \setminus D$. Then $U$ is noetherian, but $\Gamma(U) \cong \{f \in k[x,y] : f(x,0)=f(0,0)\}$ is not noetherian.