Only the Crystal Tower raids have to be completed, due to the tower itself and a certain character connected to them playing an extremely important role in the current expansion (Shadowbringers) main storyline, which SE could no longer handwave away if the player had not done those raids, so they were forced to mandate their completion by the 5.3 patch for the sake of story consistancy.

Luckily it is the only raid required to complete the main scenario, all other raids are optional, although completing the Binding Coil does give some extra lines of dialogue at the end of ARR and Heavensward's story, and the Return to Ivalice raids are required if you wish to unlock the Bozjan Resistance content to gain current end-game relic weapons ('Save the Queen'/Resistance weapons).

Everything else is optional and pretty much just fanservice fluff in some instances (*cough* the NieR crossover raids *cough cough*).