Why shouldn't variable pitch propellers be operated at low speed?
The specific reasons may be specific to each propeller/engine combination, but in general it seems to be related to stresses.

According to this King Air POH on page 2-2:


  Stabilized ground operation between 400 and 1,150 propeller RPM is prohibited. Operation between 0 and 400 propeller RPM is permissible only with propeller feathered.
  
  STABILIZED GROUND OPERATION WITHIN THE PROPELLER RESTRICTED RPM RANGE CAN GENERATE HIGH PROPELLER STRESSES AND RESULT IN PROPELLER FAILURE AND LOSS OF CONTROL OF THE AIRCRAFT.


This is probably due to the frequency of the aerodynamic forces at those RPM's resonating with the frequency of the propeller, and getting amplified to unsafe stress levels. This is called ground resonance and is also an issue with helicopters. In addition to over-stressing the propeller blades, it can also affect the fatigue life.

There is also a discussion in this thread, and it is mentioned in this guide.

As voretaq7 mentions, fixed-pitch propellers have similar restrictions for similar reasons. Some engines include a damper that removes this restriction by damping out the resonance.