Example of a compact Kähler manifold with non-finitely generated canonical ring?

As pointed out by Ruadhai Dervan in the comments, a paper by Fujino contains the answer to this question: the canonical ring of any compact Kähler manifold is finitely generated. By bimeromorphic invariance of this ring, the result even holds for compact complex manifolds in Fujiki's class $\mathcal{C}$.

The idea is to consider the Iitaka fibration of the manifold, which has the obvious property that its base is always a projective variety. Thanks to Fujino-Mori finite generation upstairs can be deduced from finite generation downstairs (with a boundary divisor term), and this latter statement follows from BCHM. The details are in the paper of Fujino cited above.

Most likely the canonical ring in Kahler situation is also finitely generated. You can check the paper http://arxiv.org/abs/1304.4013 "Minimal models for Kaehler threefolds" (Andreas Hoering, Thomas Peternell) where the MMP for Kaehler threefolds is constructed. I would expect that their results would easily imply that the canonical ring in dimension=3 is finitely generated.