This paper considers a remote state estimation problem where a sensor observes a dynamical process, and transmits local state estimates over an independent and identically distributed (i.i.d.) packet dropping channel to a remote estimator. The sensor is equipped with energy harvesting capabilities. At every discrete time instant, provided there is enough battery energy, the sensor decides whether it should transmit or not, in order to minimize the expected estimation error covariance at the remote estimator. For transmission schedules dependent only on the estimation error covariance at the remote estimator, the energy available at the sensor, and the harvested energy, we establish structural results on the optimal scheduling which show that for a given battery energy level and a given harvested energy, the optimal policy is a threshold policy on the error covariance, i.e. transmit if and only if the error covariance exceeds a certain threshold. Similarly, for a given error covariance and a given harvested energy, the optimal policy is a threshold policy on the battery level. Numerical studies confirm the qualitative behaviour predicted by our structural results.