Is there any way that the Arduino board can get damaged by using a buzzer without a resistor?
Given this common buzzer, it has a coil resistance of about 42 ohms running at 5V, it would try to draw 119mA if connected directly, far more than the 40mA (per pin) the Arduino can supply.  Drawing this current could damage the Arduino.  Adding an inline 100ohm resistor could drop the current draw to about 35ohms, within safe limits.

Also, as a magnetic component, you should place a diode across it to prevent back EMF from damaging the Arduino.  Sound vibrations can cause the buzzer to generate a voltage and the diode can prevent that from damaging the Arduino.