So I am looking at the description of the 5th level spell Animate Objects, it says that its duration is up to a minute, however it also says that, "once given an order, the creature continues to follow it until its task is complete". If the task lasts longer than a minute, does it continue to stay animated?
An animated object is only a creature while the spell is active
The spell says (emphasis added):
Each target animates and becomes a creature under your control until the spell ends or until reduced to 0 Hit Points.
As soon as the spell ends for any reason (end of duration, loss of concentration, dispelled, etc.), an object animated by the spell ceases to be a creature and reverts to being an object. At that point, any orders you have issued to the creature become irrelevant, because the creature no longer exists.