I guess the dust of disappearance is a 1 charge use-activated item that cast the spell "improved invisibility". I would treat it as if a spell caster of the level of creator of the dust had cast "imporved invisibility" on the recipient.
Therefore, if the dispel works, the character turns visible.
If the character runs into an antimagic field, the magic is supressed until the invisible person runs out of it again.
Dispel Magic: The dust is a magic item, albeit a temporary one. You would have to target a dispel on the dust to suppress it for 1d4 rounds. Since you can't see the dust, there probably isn't a way to do this.
Antimagic: Suppresses the efect of the Dust as long as the target is in the field.