Let me just give some pointers to further literature.

In the curve case, there is a cohomological obstruction to a curve over $\mathbb{Q}$ having infinitely many points, it is given by Faltings' solution of the Mordell conjecture (I know that this is almost certainly a deliberate misinterpretation of the question). In another direction, Grothendieck's section conjecture sets up a conjectural relation between rational points on curves and homotopical data, namely the etale fundamental group resp. the fundamental exact sequence.

In higher dimensions, as mentioned, the Brauer-Manin obstruction is the best known cohomological obstruction. There are refinements using non-abelian cohomology, a textbook treatment of which can be found in the book

- A. Skorobogatov. Torsors and rational points. Cambridge Tracts in Mathematics, 144. Cambridge University Press, Cambridge, 2001.

Again, in a more homotopical direction (in a way generalizing both the Brauer-Manin obstruction and the above-mentioned section conjecture stuff), there is the work of Harpaz and Schlank on homotopy obstructions to rational points.

Finally, in case the variety actually has infinitely many points, I think the Manin conjecture should also be mentioned. It provides an asymptotic of the number of points of bounded height, but (as far as I remember) the numbers influencing the growth asymptotic are essentially cohomological in nature.

If you use the keywords provided to search, you will likely end up with a number of relevant interesting papers relating rational points on varieties to cohomological of homotopical information.