If an object is rotated on its axis near the speed of light would its mass increase?
Normally if the object was moving (in relationship to the Earth for example) I would agree that its mass would increase. I think, when rotating that it won't increase mass, but I can't prove it.
In the other hand if $E=mc^2$ and $E$ is the energy of motion, does that mean that rotation has energy of motion?
Yes, this is certainly true. Mass is defined by $m^2=E^2-p^2$ (in units with $c=1$), where $(E,p)$ is the momentum four-vector built out of the mass-energy and momentum. (This defines what's known as invariant mass, as opposed to "relativistic mass.") Mass as defined in this way is not additive, and depends on the motion of the particles within a system.
As a simple example, say we have two masses $m$ at the ends of a massless stick. When the stick is at rest and not rotating, the momentum four-vectors are both $(m,0)$, the sum is $(2m,0)$, and the mass of the system is $2m$.
Now let the stick rotate end over end. The momentum vectors are now $(m\gamma,m\gamma v)$ and $(m\gamma,-m\gamma v)$. The total momentum four-vector is $(2m\gamma,0)$, which means the mass of the system is $2m\gamma$.
