I imagine you want to do is use a reverse opto-isolator or relay. Normally you have the Arduino connected to the control side of the opto-isolator, so that when the Arduino applies power (through digitalWrite), it closes the circuit and enables the remote device. Instead you want to have the doorbell circuit act as the control and it connects a circuit between a digital pin and ground. I would check whether the doorbell is A/C or D/C, and buy a relay/opto-isolator that will work with the doorbell.