Edit: This is just a long comment.

Let $X$ be a smooth projective variety.

If $\omega_X$ is ample, then $\mathrm{Aut}(X)$ is finite, so that $G$ is finite. So you see that we should avoid $\omega_X$ being ample.

If $\omega_X^\vee$ is ample, then $\mathrm{Aut}(X)$ is not necessarily finite (e.g., $X=\mathbb P^n$), but it's (the $\mathbb C$-points of) an affine finite type group scheme. In particular, its group of connected components is finite. In particular, $G$ is finite here as well.

So, as you're looking for a variety with $G$ infinite, and we should clearly avoid $\omega_X$ being ample or anti-ample, my guess is that looking at CY-variety might work. [Edit: I now realize that this never works. If $X$ is a CY manifold then the kernel is finite.]

Now, as you know, if $X$ is a curve of genus one (resp. a K3 surface), your representation $\rho$ will have finite kernel (resp. trivial kernel). So, we will have to start considering CY-threefolds.

Now, there are examples of CY-threefolds with infinite automorphism group. The first example that comes to mind is provided by the variety "X" in http://arxiv.org/pdf/1306.1590v3.pdf (also mentioned previously on MO: Can a rigid CY threefold have infinitely many automorphisms).

EDIT (thanks for David Speyer for the comments):

My first guess would be that the associated representation has infinite kernel in this case. Unfortunately, it does not. In fact, you will also have to look further than Calabi-Yau manifolds, as Prop. 2.4 in http://arxiv.org/pdf/1206.1649v3.pdf shows that the kernel of $\rho$ is always finite in this case as well.