An expert opinion on controversial video appearing to defy Newton's Laws I really want another answer to this question than just my own. 
As many people probably know the late Prof Laithwaite in the UK (a brilliant electrical engineer) made the error of wandering outside his field of expertise and claiming Newton's Laws needed amending to deal with rotating objects. 
He presents a whole series of experiments on gyros, that as far as I can see, perfectly follow Newton's Laws. There is just one video that totally baffles me. I can only think there is either intentional or unintentional deception by Alex Jones, who is demonstrating the device; see the video Patent DE2341245A1 — Vortrieb durch Präzession.
I believe what is going on is that Jones (either intentionally or otherwise) is applying a horizontal force during the initial motion. The gyro will, of course, be applying a torque, but it will be a pure torque with no net force.
Question: I know this sounds dumb, but have I got that right, or are there any alternative explanations? Is it the opinion of others seeing this video that the explanation is simply deception (as I believe it is)?
I admit to feeling a little embarrassed asking this, but assuming deception, it is fairly well disguised.
 A: Interesting. However narrator interpretation is totally wrong. Car moves because of reaction force which tries to compensate net force produced by gyroscope + arm system movement. Car movement stops when gyroscope+arm movement kinetic energy is exhausted. 
To make things more simple, you can take into account a similar system - a ball wobbling on swings, which are established onto platform with wheels. Schematics :
 
Tension + weight generates net force $F_A$ on ball, which according to third Newton law exerts same magnitude but opposite direction force $F_B$ to it's support :
$$ \vec{F}_A = -\vec{F}_B $$ Because support is coupled with platform which has wheels on it - it starts to move in $F_B$ direction. It's simple as that.
Of course gyroscope+arm system is more complex that this setup, but I am sure that basic principle is the same.
Conclusion - No any magic here and all Newton laws works as expected.
A: Just for completenes as @knzhou 's comments answer the question.
First and second laws refer to inertial frames:

1st: In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

Italics mine.

An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.
Hence, with respect to an inertial frame, an object or body accelerates only when a physical force is applied, and (following Newton's first law of motion), in the absence of a net force, a body at rest will remain at rest and a body in motion will continue to move uniformly—that is, in a straight line and at constant speed. Newtonian inertial frames transform among each other according to the Galilean group of symmetries.

Note that we are talking about laws. Laws are the axioms, distilled from data and observations, that physics theories use to pick up the mathematical model that is appropriate for the physical system under study.
The system described as a whole is not inertial, as motors have accelerating forces. In the initial inertial frame of the table the motion is defying no laws, because a force is exerted from the rotation of the motor.
All motors produce force , because of rotation, centripetal and centrifugal,  whether electric or not, and depending on the set up can induce directional motion because once a rotation starts there is no longer an inertial system for Newton's laws to hold, and in the original inertial system a force appears, which according to the first law, can change the state of the body . As khnzu states in his comemnt, the rotational force creates a frictional force in the original inertial frame of the table.
Think of an electrical motor drone, it flies in the direction chosen by the design, energy supplied by the batteries  electromotive force rotating the motor, the displacement of air (first law) producing the motion forward. ( same as with  cars and friction of the wheels on the road).
So as knzhou says in the comments to the question, if there were no friction to the wheels of the device, the system would not move.
A: I don't think that the demonstrator is giving the contraption a shove.
Suggesting that the demonstrator gives a shove is not necessary.
Let me call the direction in which the four wheels are pointing 'forward' and 'backward'.
The contraption has a bottom plate (to which the four wheels are attached) and an attached vertical plate. Then there is a movable vertical plate, the connection allows the movable plate to swing from side to side. Finally, a movable arm is connected to the movable vertical plate.
I'm spelling out all of this because I need names. From here on I will refer to 'the movable plate', and 'the movable arm'.
I will refer to the freedom of motion of the the movable arm as motion with respect to the movable plate. That is, even if the movable plate is swung sideways to 90 degrees, I will still refer to the motion of the movable arm as up/down.
Let me first discuss what the demonstrator would have to set up if the gyro wheel is not spinning.
Then he can simply lift the movable arm, and when he releases the movable arm the contraption as a whole will start to move, simply because the weight at the end of the movable arm is swinging down. 
Now to the actual demonstration, with a spinning gyro wheel.
In the starting position the demonstrator turns the movable plate sideways, but the movable arm isn't moved with respect to the movable plate.
The demonstrator releases the contraption.
The gyro wheel is in a high position so it starts to fall down. 
If the contraption as whole would not be on wheels but on an air table then the contraption would not be in a position to exchange momentum with the outside world. On an air table the contraption would mainly shift from side to side, with its center of mass remaining in the same position.
On the wheels: since the contraption as a whole won't move sideways the swing of the movable plate is relatively larger.
As the movable plate swings down the axis of the spinning gyro wheel is forced to change orientation. That forced change of orientation causes the movable arm to swing up (gyroscopic precession). As the movable arm swings down again the contraption as a whole is dragged forward by the weight of the down-moving arm.
By the looks of it:
I do see that the downswing of the movable plate overshoots the lowest point. But I think by that time the movable arm is back to resting against the movable plate. From that point on the only direction the movable arm can swing is away from the movable plate, or back to resting against the movable plate.
In the demonstration you see that for each run the system is freshly prepared.
The movable plate is lifted sideways, then the contraption is released.
The case is really not different from what you would see if the gyro wheel is not spinning. It's just that if the gyro wheel is not spinning you go straight to lifting the movable arm.
When the gyro wheel is spinning you add the intermediate step of lifting the movable plate sideways, and then the gyroscopic precession lifts the movable arm. 

A closer look at the case of the contraption suspended on an air table:
Even if the contraption is suspended on an air table it would still exchange momentum with the outside world when the sideways lifted movable plate is released. When the movable plate is released, and it starts to swing, the air suspension needs to exert a net torque to keep the bottom plate of the contraption in the horizontal orientation. So that is still exchange of momentum with the outside world.
