In case anyone specifically wants to know about the raid tier that goes live in the coming patch, in order to participate in the story version you must
  1. Complete Endwalker base MSQ
  2. Complete the Asphodelos story quests and raids
  3. Complete the Abyssos story quests and raids

If the goal is to participate in the next savage tier as soon as it releases, you must
  1. Complete Endwalker base MSQ
  2. Complete the Asphodelos story quests and raids
  3. Complete the Abyssos story quests and raids
  4. Complete the new raid tier story quests and raids

You do not need to complete the savage versions of the previous tiers, though it would be a good idea to do them for the experience.