I have a closet that contains furnace, water heater and water softener. I have a constant loud dripping noise coming from one of the three.

I had the furnace serviced and the dripping noise is still there.

Should I have the water heater serviced? Or the water softener? Is there any way for me to determine which one is the culprit?

You say you have a dripping noise... do you actually have water pooling up someoplace?