Count Olaf (as Stephano) makes the tickets to Peru appear using a magic trick. Neil Patrick Harris is a trained magician and usually finds a way to work it into his acting roles.
The Henchperson of Indeterminate Gender at one point calls themself "Doctor Lucafont," before correcting themself and saying, "Nurse Lucafont." That's because it's "Doctor Lucafont" in the book.
Olaf's suitcase contains the scarf from his Yessica Haircut disguise.
Stephano (Olaf's disguise) and The Prospero (the ship he boards) are references to The Tempest by William Shakespeare.
The base of the 'statue' in the labyrinth says, "In Loco Parentis" and "Memento Mori." In Loco Parentis is the term used for Olaf's legal guardianship of the Baudelaires. Memento Mori is a reference to the Prufrock Preparatory School motto, from The Austere Academy.