If you are asking if a plane can be taken from the gate all the way to the sky by auto-pilot I'm quite certain the answer is no. Not because a plane can't configure itself for take off and power up the engines (I'm quite certain it can, though I would defer to more knowledgeable people for that). Not for that reason, but rather because there is no system in place to electronically guide an aircraft through a series of taxiways all the way to the base of the runway so that it can line up to take off in the first place.
Keep in mind, ATC gives a lot of instructions for ground operations, funnelling aircraft this way and that via taxiway A through Z. They'd have no way of telling an auto-pilot about these directions, there's no computer to send out signals and there is no voice recognition build into the plane. If you told the pilots, they would have no way to input the instructions into the autopilot, nor would the autopilot know what to do if it ever received such instructions. It just isn't designed to follow taxiways.
Also, beyond just taxiway layout, there are a lot of planes in close proximity on the ground, plus bunches of other important vehicles (fire trucks, fuel trucks, de-icers, etc.) Steering around all of those, faultlessly, would be quite the feat. We've only just gotten to the point where a vehicle can drive across open terrain without crashing itself, I think a plane navigating an airport (and doing it while minding ATC) is a bit of a ways off.
EDIT: Raidri's video does a much better job of showing the current state of driver less vehicles, and his comment is correct. I did underestimate how good the systems are these days.
That being said, I'm not changing my answer. There still, currently, is no way to convey ATC instructions to the aircraft.
As a point of clarification: I saw the question as asking about technology in it's current state. Will we eventually be able to do this. Heck yes, absolutely. A great deal of the technology is in place and what's left is either right on the horizon or close to it. FAA clearance will probably take some time, but we'll get there. I was mainly answer the question from the perspective of modern tech and what is currently ready to be installed on a plane.