Looking at the Amazon picture of that relay, it uses an Omron G3MC-202P solid-state relay
http://www.farnell.com/datasheets/1714700.pdf
That device is for AC only, so it'll be a triac that only switches off when the current reduces to almost zero.
I couldn't see any spec for the motor control - the Craftsman PDF seems not to have the details - but it probably uses a low DC voltage (bell-wire indicates low voltage, and DC is a guess)
What appears to be happening is that when the Pi fires the relay, the triac switches on. Current flows through the triac rather than the bulb in the push-button, so that goes out, and enough current flows to keep the triac latched on.
When you press the button, you short out the triac. Current flows through the button rather than the triac, the triac unlatches, and everything goes back to square one.
Several things you can do, all of which involve not using that relay.
The easiest is to replace it with a mechanical relay.
Although solid-state relays have advantages, in this case mechanical noise isn't a problem compared to the noise of the opening door.
Electrical noise isn't either, since the opener is designed to be used with a poxy cheap little push button.
I've seen (admittedly expensive) mechanical relays operating once every two seconds for a year or so before requiring changing. I doubt you are going to operate your door often enough to reach the relay's MTBF in your lifetime.