Does leaving the Central AC on all day and all night cause the motor to burn or compressor line to freeze up?
With a properly functioning system no the motor will not burn up. And the coils and line set may show some frost but would not freeze up.

If you leave the system running and there is a drastic temp drop things might ice up a little but the throttle valve will start restricting the flow of refrigerant so the compressor may start cycling , systems usually have a minimum off time so the cycling will not over heat the motor. For systems with simple orifices the pressure switch turns off the compressor until time to start again. 
Most all residential compressors have internal thermal overloads that stop the motor from operating if they get two hot. This is yet another safety so no a properly operating hvac system will not burn up if left on all the time.