  A Laurent polynomial $f$ in two variables naturally describes a projective
curve $C(f)$ on a toric surface. We show that if $C(f)$ is a smooth curve of
genus at least 7, then $C(f)$ is not Brill-Noether general. To accomplish this,
we classify all Newton polygons that admit such curves whose divisors all have
nonnegative Brill-Noether number.
