What happens if the Moon moves twice as fast? If the Moon would rotate around the Earth with double its current speed.
What would happen then?
Would it get further from the Earth or it would be possible to keep its distance from Earth?
No.  The orbital speed is related to the distance of the moon by the equation
$$v=\sqrt{\frac{GM}{r}}$$
Where G is a constant, M is the mass of the Earth (adjusted for the mass of the moon) and r is the average distance of the moon(wikipedia).
So there are two interpretations of your question: firstly what would happen if we gave the moon a big push that doubled its speed? Well this would cause the moon to change its orbit from a roughly circular to an open orbit.  The moon would sail off into space (slowing down as it is dragged back by the Earth, and end up orbiting the sun). The escape velocity is $\sqrt2$ times the orbital velocity, so if you are in orbit, and double the speed, you are sure to escape.
If you gave it a more gentle push it would enter an elliptical orbit.  The speed would then be non-constant. If you increased its speed by 10% it would move 10% faster at perigee, but slower at apogee and on average its speed would actually be less than before.
The only way to increase the speed is to decrease the distance to the moon.  If you reduced the distance by a factor of 4 (to about 95000km).  This would cause the moon to double its speed.