I swear I'm going to make a thread that lists all of the references that my husband, myself and friends have recognized over the years that probably fly over the heads of most people. Most F.A.T.E.s and quest titles are references. Achievements, too. This is not even counting puns/wordplay (like Special Tarasque Force).
  • Sharknado - FATE -B rated movie of the same name; has a cult following.
  • By the Horns - FATE - Chocobo named "Bull Chocobo"- in reference to the phrase, "to catch a bull by the horns."
  • The Migas Touch - FATE - in reference to The Midas Touch
  • You're Tearing Me Apart - achievement - in reference to a line spoken in the B rated movie, The Room.
  • The Lady in Red - Stormblood MSQ - song reference to Lady in Red; the achievement is "Cheek to Cheek" upon completing this quest, which is a lyric from the song (in case you doubted the reference).

There's just soooooooooooo many references. I know there's a list floating around of just references made in FFXIV toward other FF titles.