Unplugging Alarm System
The system is designed to dial out on a phone line to a central station or a police monitoring system. These alarms have built in warnings for a failure in the phone line. This is a safeguard so you know the system is not working properly.

So long as the system is operational and cut off from the phone line, it will continue to beep. By removing parts of the keypad and circuit board, you have apparently shut down at least part of the system. If you truly don't want the system on, you need to remove power from the entire system. 

The power to the system is probably controlled by master box (probably where the battery was) that has a wire running to a transformer on or near your main power panel. The simplest way to stop the system is to disconnect the wires going from the transformer to the alarm control panel. 

The transformer wires are low voltage (usually 24 volts) on the outside (inside the panel they are 120 volts). To be safe, it is better to turn off the breaker connected to the transformer before disconnecting the wire. If you cant figure out which it is, have a flashlight handy, turn off the main breaker, then disconnect the wires. Then turn the main back on.

Once the system is disconnected form both the mains and the battery, it cannot send a false alarm or activate the safety beeper.