Let's try again.
The theorem of Otto and Villani implies that every distribution $\nu$ which is log-concave in the sense you define satisfies a transportation-cost inequality.
There are many distributions $\mu$ with finite moments and density supported everywhere which do not satisfy a transportation-cost inequality. Since a TCI in particular implies subgaussian concentration, any distribution $\mu$ which has larger than Gaussian tails will fail to satisfy a TCI. One such distribution $\mu$ is the exponential distribution. But by the theorem of Otto and Villani, there is no such distribution $\mu$ which is also log-concave in the sense you define.
Whichever question you mean to ask, the answer is in the above two paragraphs.