I think that the best result in this direction is the following result of Bogomolov:
Theorem Let $X$ S$ be a surface of general type with $c_1^2(X) c_1^2(S) > c_2(X)$c_2(S)$. Thern for any $g$ the curves of geometric genus $g$ on $X$ S$ form a bounded family.
In particular, since a surface of general type cannot be covered by rational or elliptic curves, these curves cannot deform. So Bogomolov's result implies that if $c_1^2(X) c_1^2(S) > c_2(X)$ c_2(S)$ then $X$ S$ contains only finitely many rational or elliptic curves.
In general, it is conjectured than rational curves are never Zariski dense on a variety $V$ of general type, and more precisely it is expected that they are contained in a proper subvariety (hyperbolicity conjecture).
If $\dim X V \geq 3$, you can obviously have infinitely many of them: for instance, take $X= Y \times V= S\times C$, where $Y$ S$ is a surface of general type containing a smooth rational curve and $C$ is a curve of genus at least $2$.
