I'm asking for an example of a finitely generated amenable group $G$ and a field $K$, such that the group ring $K[G]$ is not Noetherian.
Is it also possible to find a finitely generated amenable group $G$ and a field $K$ such that $K[G]$ does not even embed in a Noetherian ring?
For your first question, see my answer to this question. It says that if the group $G$ is not noetherian (in the sense that every subgroup is f.g.) then $K[G]$ is not noetherian for any field $K$. Now pick, for instance, any non-virtually-polycyclic f.g. solvable group, such as solvable Baumslag-Solitar groups $BS(1,n)$ for $|n|\ge 2$, or a lamplighter group, to get an example.
The second question sounds more tricky. Indeed every Ore domain embeds into a division ring (this is noetherian!) and there are many torsion-free groups for which the group algebra is known to be an Ore domain. These include poly-(torsion-free abelian) groups, for instance the solvable Baumslag-Solitar groups. This does not answer your question, but this leads me to wonder if it could not be true that every group ring embeds into a noetherian ring (of course, not a division ring!).