In The Pajama Game, the other girls in the factory are shipping Babe and Sid. Babe sings a song, "I'm Not At All In Love", in an attempt to get them to buzz off, but she and Sid end up together anyway.
Elwood P. Dowd and Harvey help Dr. Sanderson and Ms. Kelly rediscover their long-buried feelings for one another by the end of the play/film.
Crystal: So dump the chump, get another guy and let him protect you.
Chiffon: And we got one all picked out.
Ronette: A little botanical genius.
Crystal: And she ain't talkin' about George Washington Carver.
Much Ado About Nothing runs on this trope. Most of the cast ships Benedick/Beatrice and schemes to get them together. Beatrice also ships Claudio/Hero, and goes into a murderous rage when her ship gets temporarily sunk.
In A Midsummer Night's Dream, Lysander and Hermia ship Demetrius/Helena, partly because they like Helena and partly because Demetrius's plan to marry Hermia is getting in the way of their relationship. (It doesn't help that Hermia's father has talked Theseus, Duke of Athens, into supporting his preferred ship of Demetrius/Hermia.) Oberon also ships Demetrius/Helena and attempts to use love potion to make it canon. When a mix-up makes both Lysander and Demetrius love Helena, they instantly begin to ship each other with Hermia. Finally, Oberon fixes everything and his two favorite couples become Ship Mates. Even Theseus gets on board in the end.
The subplot in The Taming of the Shrew mostly exists because Lucentio's servant, Tranio, ships Lucentio/Bianca. He eggs his master on in pursuit of the girl, then instigates a Zany Scheme to make it happen, eliminating her other suitors while he's at it.
Deconstructed in Cyrano de Bergerac: Christian forces Cyrano to tell the truth to Roxanne because he wants Roxanne to choose him instead of Cyrano.
In Hamlet, Queen Gertrude jumps pretty quickly to the conclusion that Hamlet's madness is caused by his love for Ophelia. She says several times that she hopes it to be the case and, at Ophelia's funeral, says she wishes Ophelia had been the girl Hamlet married.