As soon as you complete all the quests on an island, you will be able to move to the next one. When you reach to the next island, a timer will start on the previous one (the island you just completed). This timer is 24 hours long and within this time you can still access to this island to collect resources, treasures etc.
When the 24 hour timer is over, you will no longer have access to the island. The reason for that is to optimize the game so everyone can play without having connection or memory problems. There's no need to worry about the closed islands though. When you complete all the quests on an island, it means all quests related to the main storyline are done.