The safest thing would be a relay. I assume your garage door opener has a wired/manual momentary pushbutton switch? A relay is just an electrically-isolated electrically-operated switch and you can wire the normally-open relay contacts in parallel with the (normally open) switch contacts. Then you have "or logic" where it can be operated with the relay or the switch.
You will need a driver for the relay coil.
Or, [u]here[/u] is an example of a transistor being used as "pull down" switch. You can probably leave out RL, since you say the voltage is already pulled-up to 5V. Rin can be somewhere around 1K. Any NPN transistor should work since you'll have low voltage and (hopefully) low current. The transistor will invert the signal so writing a logic-high will turn-on the transistor, pulling the collector to ground.