No, the Farm Innovators Bird Bath De-Icer Aluminum (C50) will not shut off on its own if the bird bath is empty. Furthermore, it's important to keep an eye on the water level of the bird bath: if it runs out and the De-Icer is left running without water, it could burn out.
The Farm Innovators Bird Bath De - Icer Aluminum is designed to be completely submerged in any birdbath. The heating unit and the cord attached are designed to be submerged in water so that the unit can sense when it needs to turn on.
The Farm Innovators Bird Bath De - Icer Aluminum is designed to be completely submerged in any birdbath, so any leaves it could contact there would not catch fire as they would be too wet. It should not be left out of water as it can burn out.