Activate garage remote via an Arduino
Connecting arduino ground and an IO pin across the button is probably really not recommended. For a start you might find that your arduino ground and IO pin are now forcing the transmitter to have maybe up to 6V across it: -







simulate this circuit &ndash; Schematic created using CircuitLab

I'm not saying the wiring is like this BUT neither are you - you are guessing and you might do some damage so tread carefully. Ideally you need to connect the arduino ground pin to ground on the radio (see the symbol I've drawn) and the GPIO line to where the transmit circuit of the switch is.

I can also see why your battery became drained because, when you shut your arduino down it in effect shorted the switch and probably transmitted a constant carrier wave all day!!