Air purifier keeps making a noise even after we cleaned it several times and left it out to dry several hours.

Did you clean the blades, and let them dry 24 hours?

The clicking noise is the sound of the high ionizing voltage 'shorting' due to dirt or humidity or both which provides the pathway for discharge. Cleaning and completely drying the filter(s) does not always help, in this case switching the unit to a lower operational speed can mitigate this problem. New filter(s) may be needed to resolve the clicking.

try cleaning the grid blades with a scotch brite pad and clean the corona wires very gently with light grade sandpaper. this worked on one of my units.

The filters can be cleaned with a vaum. Then hold the filter button down for 60 seconds and the light wil go off.

I also have this problem as soon as I pt a new filter in - does anyone have a clue what this problem is?

after cleaning is it dry. you might want to take apart and see what is clicking wheel or some thing hard to tell since i can't see it. as far as options . check with the manufacture google it check them out on line hope this helps. Tom

You are probably aware that the ionizer uses a very high voltage (at very low current) to function. If you have used a cleaner of some kind on it, you could have made the grid sensitive to moisture.
I would remove the grid again and wash it thoroughly in distilled water to remove any residue that stayed behind, then dry it in warm air before reinstalling it. 

there's usually a troubleshooting guide stuck on the inside of the door.  it suggests a systematic flowchart for diagnosing the problem.
you should be able to isolate it to either the power supply (means replacement), or one of the cells.
on mine, one of the many little vanes got a bent during cleaning, and was shorting out.  (it doesn't take much.)  a quick visual and very little judiciously applied force straightened it out and fixed the problem.
i've also found that the unit will crackle a lot after a cleaning because of drops of water remaining in the cells, so i usually reassemble everything, but leave the unit turned off for an hour or two, which lets the blower on the furnace dry the cells for me.

clean your cell. Allso check for broken or loose tungsten wires in cell.