How can removing a piezoelectric speaker/buzzer damage a circuit?
Modifying a circuit that you don't understand can be dangerous, so it pays to be cautious. Without knowing all the design parameters you can't be sure that your modifications won't have unintended consequences.

Piezo speakers look like a capacitor to the driving circuit, typically 10~100nF for the elements used in piezo buzzers. If the driver uses a transformer then it may rely on this capacitance to suppress high voltage spikes and/or 'tune' the oscillation to the correct operating frequency. Without a speaker it may produce higher voltage and/or frequency, which could cause interference or even damage the transformer or other components. 

However most piezo buzzer circuits do not use a transformer, so disconnecting the speaker won't cause them any harm. Trace the wires from the buzzer back to the driver circuit. If you don't see a transformer there then it should be safe to disconnect the buzzer.