First of all, the buzzer is there for compliance reasons (sound alarm is required for automated garage door controller), and if you build it DIY you don’t necessarily have to use the buzzer.
Second, you should check the buzzer’s spec — is it an active low or active high type? The OpenGarage circuit uses an NPN transistor to switch the buzzer, and it’s active high (which means a high signal turns on the buzzer and low signal turns it off). If your buzzer is the reverse (some buzzer modules are active low type, because they use a PNP transistor), then the logic is reversed, and that explains why it’s buzzing continuously.