I mocked up the circuit that eliminates the popping sound heard when the mic button is pressed on a mic cable. However, the signal that the capacitor is attached to, the other terminal to R || Switch to GND, does not go low even when I hold the button down. Essentially, the mic is not muting.
You want a high resistance for the resistor - it is just there to ensure that the capacitor is charged to the average DC voltage on the audio line, so there won't be a "pop" when you close the switch to connect the bottom of the capacitor to ground.
The capacitor must be a fairly high value to present a very low impedance at audio frequencies when the switch is closed. The low impedance of the capacitor effectively shorts the audio signal to ground with the switch closed.
If you don't hear any effect on the audio level when you close the switch, you probably have something wired incorrectly.