I found several examples where the infinitesimal Torelli theorem (e.g. as stated by Carlson, Green, Griffiths, and Harris on p. 144 of ``Infinitesimal variations of Hodge structure I") holds in the literature, but it was strange that I couldn't find anything worked out explicitly about Jacobians or abelian varieties in general without resorting to the more general case of smooth projective varieties with trivial canonical bundle (although these cases are known as mentioned in the comments).

In particular, I tried looking at the cup product map associated with the theta divisor of the Jacobian. More specifically, we consider how $H^{1, 1}$ varies when $J(C)$ is deformed as $C$ varies in a family of curves with a fixed automorphism group $G$. If the infinitesimal Torelli theorem holds, then this map must be injective (i.e. the dual map is surjective). At first, this seemed like something doable since the evaluation of the dual map at a fixed point is something we can write down easily in terms of linear algebra (which is known very explicitly in specific cases -- see ``On the Neron-Severi group of Jacobians of curves with automorphisms'' by Giovanetti).

However, actually checking when surjectivity holds in general even for this specific component seemed a lot more complicated than I expected without making assumptions on the point (which is actually a matrix) such as diagonalizability. Is there something I'm missing in my (attempted) computations or is there some deeper problem that I'm not aware of?

EDIT: Here are some updates/some sources of confusion:

According to p. 95 of ``Algebraic Geometry III'' by Parshin and Shafarevich, Griffiths proves a condition which can be used to show that the existence of a deformation of a principally polarized abelian variety $X$ giving rise to an injective Kodaira-Spencer map (i.e. an effective deformation) implies that the infinitesimal Torelli theorem for $X$ (the argument outlined seems to assume the deformation is effective). 

In the case that $C$ is a non-hyperelliptic curve of genus $g \ge 3$, the map $H^0(K_C) \times H^0(K_C) \longrightarrow H^0(2K_C)$ is surjective by Noether's theorem. Combining this with the projection map to the invariant subspace $H^0(2K_C)^{\text{Aut}(C)}$, we  find that the map $q$ given on p. 314 of the Giovanetti paper is surjective. In the specific familiy considered in the paper, the map $q$ is the dual of the Kodaira-Spencer map, which seems to imply that the Kodaira-Spencer map is injective and that the criterion mentioned above applies. This would mean that the infinitesimal Torelli theorem holds for $J(C)$ in this case. 

Now, this seems to imply that the dual of the cup product map is surjective. However, comparing the dimensions of the spaces considered on the maps on the previous page seem to imply that the dual of the cup product map can't possibly be surjective. I think I'm either making a mistake in the assumptions/steps made above or being confused about whether I'm taking duals of various maps. What exactly is happening here?

The other thing is that the Giovanetti paper assumes the surjectivity of the Kodaira-Spencer map. If the criterion above holds, we seem to actually find that this is actually bijective. Does this indicate any potential problems so far?

Note: Thank you for the comments! As mentioned below, the infinitesimal Torelli theorem should hold for abelian varieties. However, I'm still confused about how the maps in the Giovanetti paper work out. In particular, the map that seems to correspond to the cup product maps from a vector space of higher dimension to a lower one. Finally, is there a more concrete way to prove this in the specific case of Jacobians?