Anne of Green Gables has the idea of hurrying along a couple by having the woman pretend to be dating somebody else. This forces the guy to propose. This is made all the better by the fact that the patsy brought in to pretend to court the woman hints that he fell in love with her, too.
She then attempts this strategy with another couple. It backfires because she didn't know that the guy was trapped by a promise not to marry anybody.
She even used it herself. She has rejected Gilbert, who is now seeing Christine Stuart. She's not really in the mood to go to a party, and acts rather distant on the walk there, but the very minute she enters the room she starts pretending to be the most fascinating and wonderful girl present, drawing all attention (including Gilbert's) to herself.
Mr. Rochester pretends to be in love with Rich Bitch Miss Ingram because he knows that Jane is emotionless and placid on the outside but fiery on the inside, and that jealousy will be the best way to bring her feelings for him to the surface.
Jane later snaps Rochester out of his melancholy self-pity by arousing his jealousy of St John Rivers. It enables him to finally spit out that he still loves her and wants to marry her.
Hermione dates Cormac McLaggen in Half-Blood Prince solely to make Ron jealous, and immediately regrets it. She later tells Harry that she purposely chose the person who would annoy Ron the most.
In Pottermore, Tonks made some casual remarks to Lupin about Siriusnote who just so happens to be her cousin being not-so-bad-looking. Lupin got annoyed that she seemingly fancied Sirius, and she replied that he knew perfectly well that it wasn't Sirius whom she actually fancied.
In P. G. Wodehouse' books, Bertie Wooster would often use this technique to speed up marriages, although they were never meant to be his own. He himself would play The Beard, usually without asking anyone's permission first.
Tom and Becky try to overdo each other in The Adventures of Tom Sawyer. They got engaged, but Becky breaks it over when Tom blabbled about having been engaged before. When he returns from the island as the hero, Becky refuses to reconciliate. Tom flirts with Amy and Becky suffers, but strikes back and flirts with Tom's mortal enemy Alfred Temple.
"Tom fled home at noon. His conscience could not endure any more of Amy's grateful happiness, and his jealousy could bear no more of the other distress. Becky resumed her picture inspections with Alfred, but as the minutes dragged along and no Tom came to suffer, her triumph began to cloud and she lost interest; gravity and absentmindedness followed, and then melancholy; two or three times she pricked up her ear at a footstep, but it was a false hope; no Tom came."
Kitty tries this on Jack in Georgette Heyer's Cotillion. A subversion, in fact, since it works, to a point, but by that point Kitty's in love with Freddy.
Daisy Miller: An Alternate Character Interpretation that Daisy was using Giovanelli all along to make Winterbourne jealous after she felt he let her down in Vevey, or that she decided to mess with Winterbourne's head when she saw how much her flirting with Giovanelli annoyed him, started such a heated FanWar that Henry James had to write his friend and assure her that Daisy was never planning anything of the sort; she was too clueless in the politics of romance to plan such a thing. Not that it matters, anyway...
In The Sword of Truth, Richard's former girlfriend didn't like their relationship developing too slowly. So, she let him catch her screwing his brother and invited him to join. Got a blank stare, and that was it.
In Gone with the Wind, Scarlett attempts to use other men's interest in her (Charles's, specifically) to get Ashley to break off his engagement to Melanie and propose to her instead. It doesn't work.
The Agatha Christie character Mr. Parker Pyne, who runs a business which advertises with the simple question "Are you happy?" and aims to solve any problem this might bring to him, has a male and female employee on staff who are experts at this. However, the trope never gets played quite straight in the published stories; the closest is "The Case of the Discontented Husband", where it goes horribly wrong.
In The Mortal Instruments, Simon hangs around Isabelle to make Clary jealous when it seems like she's not interested in him, although this develops into something more genuine.
Thoroughly deconstructed in Chronicles of the Kencyrath. Timmon likes Jame, and Jame kind of likes Timmon too. But she likes someone else more, and Timmon can be a dick sometimes, so she's not really interested in getting together with him. So Timmon launches Operation Jealousy and gets with Narsa. It actually does work—Jame, much to her annoyance, is a little jealous. But she's not jealous enough to actually do anything about it, and she's annoyed at Timmon for pulling this. Meanwhile, it turns out Narsa is a Clingy Jealous Girl, and furious and heartbroken over her role in all of this. She tries to kill Jame, and later, after Timmon gets her pregnant, kills herself.