In this paper, we study the optimal sensing scheduling policy for an energy harvesting sensing system equipped with a finite battery. The objective is to strategically select the sensing epochs such that the long-term average sensing performance is optimized. In the sensing system, it is assumed that the sensing performance depends on the time duration between two consecutive sensing epochs. Example applications include reconstructing a wide-sense stationary random process by using discrete-time samples collected by a sensor. The randomness of the energy harvesting process and the finite battery constraint at the sensor make the optimal sensing scheduling very challenging. Assuming the energy harvesting process is a Poisson random process, we first identify a performance limit on the long-term average sensing performance of the system without the finite battery constraint. We then propose an energy-aware adaptive sensing scheduling policy, which dynamically chooses the next sensing epoch based on the battery level at the current sensing epoch. We show that as the battery size increases, the sensing performance under the adaptive sensing policy asymptotically converges to the performance limit of the system with an infinite battery, thus it is asymptotically optimal. The convergence rate is also analytically characterized.