Channel mutes will mute everything from that channel. This includes all mix busses. The mute group is likely muting the individual channel busses, thus muting the monitors.
One solution would be to create a stereo group, and send all of the items that are part of the mute group to that stereo group and, then send that group to the LR mix. Now mute that stereo group instead of the source channels. This will mute the FOH, while still passing the audio to the monitors. Be careful of effects when doing this, make sure that the FX returns for the items being muted are also muted.