The modifier created by Voidfire Wand's triggered power prevents your opponent from drawing a card the NEXT time they would draw a card. It does not state "this turn". So whether it's the same turn you deal combat damage with the Wand or two turns down the road, they will not draw a card the next time that they would.
Magneto:We are the future, Charles; not them.
Player Management: Level 2
Rules Knowledge - VS System: Level 1
Rules Knowledge - World of Warcraft: Level 2
Rules Knowledge - Yu-Gi-Oh!: Level 2
Tournament Organizer: Level 1