It won't cause damage to connect 4xAA to the input of the Arduino's regulator, however it also won't result in specified operation, as even with fairly new cells the voltage will below that which the typically used 5v regulators require to deliver 5v at the output.
Additionally, the common L293D and L298 based bipolar motor shields are horribly lossy. Feed them about 6v input, and you'll get just over 4v to your motor, which is likely to be disappointing.
To build an Arduino-based battery-powered motor project, you typically need to do one of two things
Use 6, 7, 8 or more cells, and higher-voltage, lower current motors
Use a 3.3v Arduino and a more efficient FET-based H-bridge motor driver/shield, for example the TB6612FNG. This supports 4 cell operation quite well, and may to a more limited degree work on 3 alkaline cells.