Adding 40db or more typical power amp gain to the noise and attenuating the console output needlessly, results in less than optimum signal to noise ratio heard on the monitor loudspeakers. The residual console noise is perceived as a masking signal which degrades the transparency of the perceived mix. The mixer often adds more eq or compression than necessary to get the mix to jump out of the muck. Consequently on a playback system without the masking noise, the mix sounds overly processed and unpleasant to hear. The same principle applies to Cue systems.
Several manufacturers make such devices and they are fairly inexpensive.
It's well worth the effort. One Bit DACs and Music Several CD players today 'feature' 1 bit DACs and oversampling. This is done because a 1 bit DAC has no non linearity's, as do all multi bit DACs. The output of a 1 bit DAC is either 1 or 0 and as such can have no error. These DACs are cheaper to produce.