For general subvarieties of an algebraic torus, the tropicalization knows about the class of the subvariety in a suitable toric compactification of the algebraic torus. So you can compute intersection products of subvarieties of a torus tropically. See the recent preprint of Osserman-Payne for the state of the art.

For subvarieties X that are schon (which is a natural smoothness condition), you can say a lot more. There is a natural dualizing complex $\Gamma_X$ which maps to Trop(X) whose homology reflects the lowest bit of the weight filtration on X. From this fact, you can get the natural generalization of $g(X)\geq b_1(\Gamma)$ (it is not in general true that $g(X)\geq b_1(\operatorname{Trop}(X))$ because the tropicalization map may have disconnected fibers (as was pointed out by Speyer)).

There are two special cases where you can say a lot more: when $X$ is a schon hypersurface and when Trop(X) is smooth (smoothness here means Trop(X) is locally modeled on matroid fans). In this case, you can say things about the Hodge numbers of X. See my paper with Stapledon for details. Warning: the results of that paper require compactifying X by completing the algebraic torus to a toric variety. We have a sequel in the works that will use more sophisticated Hodge theory to get around that problem.