Does it require a torque to spin a rectangular plate around its diagonal? If a rectangular plate is rotating aroud its diagonal, one can find its inertia tensor and see that the angular momentum vector is changing in time. Thus it should require a torque to keep spinning, right?

Does this mean that if one would set it spinning around the fixed axis (its diagonal), it would eventually stop and come to rest? 
If this is true, where did the kinetic energy go?
 A: The answer is that a torque needs to be applied to the rectangular plate by the bearings of the support but that torque does no work so the kinetic energy of the plate stays constant.  
The left-hand diagram below shows the directions of the angular velocity of the palte $\vec \omega$ and its angular momentum $\vec L$ which is in the plane of the plate.  

As the plate rotates the direction of the angular momentum changes as shown in the right-hand diagram and so there must be a torque (couple) acting on the plate.
The magnitude of the angular momentum does not change.
The torque $\vec \tau$ is provided by forces $F$ which are in the plane of the plate and acting on the plate at the bearings.
The torque does no work on the plate, so the kinetic energy of the plate is constant, it only changes the angular momentum of the plate.
Each of the particle which make up the plate execute a circular orbit and so the force on each of those particles must be at right angle to their velocity which means that no work is done by the forces which are accelerating the particles and changing the angular momentum of the plate.
In terms of the external couple which is applied on the plate, that couple does no work, there is no displacement of the external forces along the line of action of the forces, rather all that happens to the external torque is that it changes its direction as the plate rotates.
