I am looking for a smooth proper curve $C$ such that there does not exist any closed embedding $C to S$ where $S$ is a (normal projective) toric surface.

Since $C$ is smooth I believe it suffices to consider smooth projective toric surfaces $S$ since we may always perform a toric resolution of singularities and the strict transform of $C$ will be isomorphic to $C$ since $C$ is smooth.

Using the result on p.25 of Harris Mumford, On the kodaira dimension of the moduli space of curves, I can conclude that a very general curve cannot have any such embedding.

However, I am not able to write down an explicit example. Does anyone know such an example or what sort of obstruction might work to check this in particular examples.