Using I/Q demodulation/LIA for a fixed-frequency application of several signals - do the phases of the input signals affect the result? My input is a single signal composed of the sum of four periodic signals, each 4 Hz apart, centered around some resonant f0 typically between 1200 - 1300 Hz; the actual frequencies will change occasionally, but not the distance between them. For example, if f0=1250, my frequencies will be [f1..f4] = 1244, 1248, 1252, and 1256 Hz.

A decent analogy would be four LEDs flashing at these frequencies, observed by a single detector.

This signal is sampled at 16384 Hz, and I perform IQ demodulation in software (described here) using the [f1..f4] as the reference frequencies to ultimately extract the amplitude of those four components from the original signal.

My question is about the phases of the four different components. I can control their phases, and I'm wondering if there are any best practices to make sure I get the cleanest possible separation of the four signals, or if the phase won't matter?
Under the assumption that your words "extract the amplitude of those four components" actually mean "extract the spectral magnitudes of those four components", then the initial phases of the original four periodic signals do not matter. It's straightforward to prove this behavior by modeling your down-conversion process using software.