Do I have to supply an analog switch IC with a voltage level higher than the switched voltage level?
The recommended operating conditions suggest your input signal should not be outside the power supply range. The absolute maximum ratings state that exceeding the power rails by more than 0.5 volts may damage the device (orange rectangle below).




  I want to know if I need to apply at least +5.4V VCC and -5.4V VEE
  voltages in order to transfer a ±5.4V signal.


Yes that would be OK


  What happens if I supply the IC with lower VCC and VEE voltages (e.g.;
  VCC = +3.3V, VEE = 0V). Does my digital signal get clamped? Does
  anything bad happens other than seeing a higher switch channel
  impedance?


You risk damaging the chip. However, if you limit the current drive capacity of your slightly excessive signal to significantly within the +/- 25 mA specified in the absolute maximum ratings, then this should be fine but, your signal will become clipped.