Will cutting power to a CMOS chip effectively remove it from the circuit?
The quick answer is, no.  It won't work.  Most IC's have ESD protection diodes on the inputs and sometimes the outputs.  If the chip is powered down, it will be like having a diode to 0 volts on the signal.  Best case, that signal will see a large load on it.  Worst case, the signal+diode will actually cause VCC to go up, causing the chip to actually power up.

Here's a schematic that shows some protection diodes.  The schematic isn't great, but it does the job.  Imagine everything to the right of the dashed line to be inside the chip:



The two diodes from ground and to VDD are there to prevent the input signal from going too much above or below the power rails.  Everything to the right of the diodes can safely be ignored, as they are chip specific.

As you can see, if the input goes above VDD (which is 0 volts when powered down) then there is a current path from the pin to VDD.