Ok I didn't make my self so clear.
De-Ionizing the air should cause the air to contain less dust, as it will be able to sustain (hold) a lower volume of dust with in it's own volume.
The dust will of course go some where, namely it will be dropped via gravity on to any horizontal surface. (you'll also get less dust sticking to vertical surfaces)
If this device has a filter in then yes it will trap dust just as any fan+filter will (and you will regularly need to clean the filter.
Just checked the website, looks like this one uses an electromagnetic collection plate, causing the dust from dust laden air moving through it to stick to the plates instead of a physical filter (you'll still need to regularly clean the collection plates)
You could try a vacuum cleaner to reduce the total volume of dust from the room