Although she may be referred to as simply "the librarian" during the first two episodes of A Series Of Unfortunate Events Season 2, Sara Rue's character has a complicated backstory. Spoilers for all of Season 2 follow. Sara Rue is credited as Olivia Caliban in A Series Of Unfortunate Events and readers of the book series will surely recognize that name even though Olivia Caliban doesn't appear anywhere in The Austere Academy. She makes her first (and only) appearance in the book series in The Carnivorous Carnival as fortune teller Madame Lulu. But Olivia gets a major upgrade in the Netflix series and it makes this V.F.D. member a far more interesting and sympathetic character.
No matter where they land, the library is always a place of refuge for the Baudelaire orphans. And although she's not mentioned in The Austere Academy, the librarian at Prufrock Preparatory School is Violet, Klaus, and Sunny's ally in Season 2 of the TV series. The Lemony Snicket wiki notes that the librarian was mentioned in Lemony Snicket: The Unauthorized Autobiography and confirmed as a member of the V.F.D., but the Netflix show decided to flesh out this character and combine her with Olivia. This Olivia joins forces with Nathan Fillion's Jacques Snicket in an attempt to save the Baudelaires and Quagmires and in the process, she becomes a volunteer. Due to these changes, Olivia is not only a more significant character in the TV show than she is in the books, but she's also a person worth trusting.
In the books, Olivia is also a member of the V.F.D., but her allegiance after the schism of the group is unclear. She doesn't show up until the ninth book and the Baudelaires don't recognize her at the Caligari Carnival because they've truly never seen her before. Just like in the TV series, Olivia pretends to be the psychic Madame Lulu and the cover protects her connection to the V.F.D. and allows her to hide a library. But unlike the Olivia in the TV series, the Olivia in the books isn't all good.
While both versions of Olivia do think there's a possibility that one of the Baudelaire parents survived the fire and agree to travel to the Mortmain Mountains with the children, their motivations are quite dissimilar. Instead of intentionally placing herself at the carnival to help the V.F.D., Olivia in the book wants to give all people what they want — and that includes Count Olaf. So even though book Olivia gives the Baudelaires some essential information about the V.F.D. (as they so desire), she also tells Count Olaf that the freaks Chabo, Beverly, and Elliot are really Sunny, Violet, and Klaus. Because, after all, there's nothing that Count Olaf wants more than to know the whereabouts of the three Baudelaires.
Another difference from the books is that the Madame Lulu alias seems to be an important element in the Netflix show. As TV Olivia explains, the fortune teller is a rotating position for whichever V.F.D. member is stationed at the carnival. That person uses the cover to collect information for the Volunteer Fire Department and before Olivia arrived, it appears that Allison Williams' mysterious character (is she Beatrice? the Baudelaires' mother? one in the same? or perhaps Kit Snicket?) was in the Madame Lulu role. So Madame Lulu will live on in Season 3.
Olivia, on the unfortunate other hand, will not. Because despite the departure from her book counterpart, Olivia's fate on the show is the same as it is in The Carnivorous Carnival. Lemony Snicket had warned that the librarian's future would be grim in the beginning of Season 2 and you should always heed his warnings. Olivia ends up being eaten by lions. While the Baudelaires were sad about her death in the books (particularly before they realized she ratted them out to Count Olaf), it's downright tragic to lose the newly-minted volunteer in the TV series.
You may still be mourning the loss of Rue, but the expansion of the Olivia character does provide insights into the V.F.D. and Jacques Snicket that were not offered directly in the books. And while you shouldn't expect to see Olivia in Season 3, her favorite quote will continue to ring in your ears, "In a world too often governed by corruption and arrogance, it can be difficult to stay true to one's philosophical and literary principles." At least Netflix's version of Olivia succeeded in doing that.