A positroid is a matroid realized by a matrix such that all maximal minors are non-negative. Positroid polytopes are matroid polytopes of positroids. In particular, they are lattice polytopes. The Ehrhart polynomial of a lattice polytope counts the number of integer points in the dilation of that polytope. The Ehrhart series is the generating function of the Ehrhart polynomial, which is a rational function with the numerator called the h^-polynomial. We compute the h^-polynomials of an arbitrary positroid polytope by a family of shelling orders of it. We also compute the h^*-polynomial of any positroid polytope with some facets removed, and we relate it to the descents of permutations. Our result generalizes that of Early, Kim, and Li for hypersimplices.