Does a CE amplifier draw more power when it is amplifying than it is sitting idle?
I'm assuming you mean a class A CE amplifier like this:


At it's idle state, a certain quiescent current flows which determines the power taken from the supply. Easy :-)

Now let's see what happens when we apply a certain input signal. I'm assuming that the input signal is small enough so that no significant distortion occurs. The input signal will be equally positive as negative as is normal for audio, I mean that it's average value will be zero. 

During the positive part of the input signal the transistor will conduct a little bit more. This causes the collector current Ic to increase proportionally to the input signal. So the circuit will consume a little bit more power.

During the negative part of the input signal the transistor will conduct a little bit less. This causes the collector current Ic to decrease proportionally to the input signal. So the circuit will consume a little bit less power.

On average, this will cancel out and the average current consumption will be equal to the quiescent power consumption.

However, things will be different depending on, for example:

How you load the output

If for some reason positive and negative do not cancel, this would mean distortion occurs.