With no real time frame, just a lot of speculation, by the time the show might need to be moved they will have launched, or be very near launching, the national tour. In that case they will have figured out other ways to do the effects and could implement that version of the show here in NY, similar to what happened with Beauty and the Beast when it moved.
And the larger majority of the time the theatre was dark between shows was more for general refurbishment and upkeep and not as much about retrofitting the stage just for the effects in Aladdin. After all, they didn't have a several months long load in for the pre-Broadway run in Toronto.
But it's all speculation on our part. Who knows what kind of shape ticket-sales will be in by the time Frozen is actually ready to start. I do think that if the show is still pulling in these same kinds of numbers on a regular basis that Disney will consider moving it before closing it, or putting Frozen into a different theatre. After all, in theory the shows would be covering different demographics (Aladdin getting the boys, Frozen covering the girls), as opposed to when they claimed Beauty and the Beast was closing for Little Mermaid because they didn't want to have two "princess" musicals running simultaneously.