Building a wireless door opener
If you are a programmer, one of the simplest way is to use an USB relay and an network enabled mini computer, such as Beaglebone Black or Raspberri PI. That way you wont have to do any electronics at all.

USB relays are very common, for instance:

http://www.phidgets.com/products.php?category=9

http://www.yoctopuce.com/EN/products/category/usb-actuators  (*)

http://sigma-shop.com/product/67/usb-relay-controller-one-channel-pcb.html

Just make sure you choose one matching or exceeding your door voltage/current requirements. Also, check what software is provided, being able to control it with proper API would certainly help :-) 

One last thing: your door control wire is probably driving a big coil, you might have to protect the relay from voltage surges, with a TVS diode for instance.

(*) I do work for the company making theses.