I have water softner located outside of house. On kitcen sink I have reverse osmosis system. Under kitchen sink is the small tank and 3 filters. Recently the plumer replaced the filters. Then the moaning began. I hear the vibration/moaning coming from under the sink and I can feel the vibration when I touch the filter.

How do I stop this annoying noise?

Sounds like the plumber left you with a loose filter. Call him back to tie everything up. This should be done under warranty at no charge to you. Good luck, Tom