Choir Members Can't Hear Themselves
Maybe the piano, organ, or tape tracks are too loud in the choir monitor speakers.
Have the choir sing a capella, with the HVAC system turned off. Can they hear themselves? Now turn on the HVAC system. Can they still hear themselves? Now turn up the piano or organ in the choir monitors. Then turn up tape tracks. At what point can the choir no longer hear themselves? Turn down the offending sound source.
If the problem persists, consider another choir monitoring strategy.

Hum or Buzzes
Power amplifiers can radiate strong hum fields, so keep other components away from them.
Keep microphone cables separate from power cables. If these cables must cross, cross them at right angles and space them vertically.
Power the audio system and from its own circuit breaker. Do not share the circuit with the lighting system.
If the cable connecting two devices has XLR connectors on both ends, unsolder the shield from XLR pin 1 in one connector. Or insert a ground-lift adapter. Do this for both channels.
Mic cables are an exception to the rule above. In each mic cable, check that the shield is soldered to XLR pin 1 on BOTH ends.
If the cable connecting two devices is unbalanced, you might wire an audio isolation transformer between them. Do this for both channels.
Check that the set screw in each mic XLR connector is screwed in tight.
In direct boxes, set the ground lift switch to the position where you monitor the least hum.

Dull, Muffled Sound
Check that the high-frequency drivers in the loudspeakers are working. They may have been blown out by amplifier clipping.
Be sure the high-frequency EQ is not turned down.
If only the playback cassette deck sounds dull, clean the heads with a cotton swab and isopropyl alcohol (available at a drug store). You might need to have the deck inspected for head alignment. Also, be sure that the noise reduction is switched off if the cassette was recorded without noise reduction.

Unnatural Tone Quality
Try better microphones.
Adjust the EQ on each microphone until the tone quality is natural.
If everything sounds unnatural, use a graphic or parametric equalizer. Or use better loudspeakers.
Be sure all the loudspeaker drivers (woofer and tweeter) are working.

Feedback
Turn down the volume on the microphone that is feeding back.
Use a parametric equalizer on the monitor speakers to remove frequencies that feed back.
Try an automatic feedback cancelling device, such as the Sabine FBX Feedback Exterminator.
Use directional microphones placed close to instruments.
Place monitor speakers at the point of least sensitivity of each microphone. This angle, relative to the front of the microphone, is shown in the polar-pattern diagram in the microphone data sheet. It is the angle where the polar-response curve goes closest to the center.
Use as few microphones as possible.
Turn off microphones not in use.
Use an automatic (gated) mic mixer.
Keep loudspeakers far from microphones.
Use directional speakers that focus the sound on the audience.
If you're miking a group of singers with a single mic, and you get feedback from the floor monitors, give each performer a closeup mic instead.
Move the mic closer to the person speaking, or ask them to talk closer to the mic. Demonstrate to them how they can't be heard if they're too far from the mic.

Bruce Bartlett is a microphone engineer for Crown International, a recording engineer, and an audio journalist.