On Deviant ART, a user known as Son Amy Fan 362 is writing an original Fan Fiction series. It's called "A New Life Era". In the first two stories, there is very strong evidence to one of the protagonists that his best friend's daughter may be crushing on someone close.
The first story of the series: http://sonamyfan362.deviantart.com/art/A-New-Life-Era-351466819 focuses on a character known only as "The Hacker", preparing for his son and daughter's birthday, is demonstrating the behavior of a matchmaker. He sees that there may be more than meets the eye between two (unrelated) children. However, everyone else seems to be in denial. However, apparently, this doesn't discourage "The Hacker" from his shipping hobby.
In The Fledgling Year, many of the girls who are supposed to be Corís potential brides donít bother trying to ingratiate themselvesónot because they donít like him, but because itís obvious to everyone but him that heís in love with Aravis. Hana particularly outright says that itís ridiculous to think of anyone but Aravis marrying Cor, and periodically gives her advice on their relationship.
Later on, Beth is an all-out shipper for Watson and Sally, realizing that Sally is the mysterious second Mrs. Watson of the Sherlockian Canon and thinking of the newlyweds as the cutest thing ever.
Sally, in turn, fully supports Beth in her love for Holmes and practically squees when she sees the two of them married and simply acting like newlyweds.
The Doctor ships both Watson/Sally and Holmes/Beth, although he's more active in encouraging the latter.
Played for Laughs in Fly Free, where Sakura is convinced that Robin and Kakashi are meant for each other. Other characters, including Ino, Naruto, and Sasuke are convinced by the 'proof' that Sakura shows them.
"Do you know how many tires I had to slash, how many boys I had to threaten, how many dates I had to turn down just to make the way clear for the two of you to finally see each other for what you were? Half the fights I got into at school in some way involved you lot. But that's ok, that's what I wanted. The two of you belong together and not just because it was necessary to maintain the timeline, complete the paradox and create me, but because you truly love each other."
In the Fallout 3 fan fiction Trouble, Johnny Saint the legendary vault dweller can see the chemistry going on between Harkness and Butch, and casually offers Harkness advice on how to deal with Butch's emotions.
Lissa and Sumia support Frederick/Robin after realizing the two are spending a lot of time together in Pretender. Justified in Sumia's case, she knows that Robin has feelings for her husband Chrom and hopes that if Robin moves on, Chrom will stop being obsessed with Robin.
Lereal Belsai, the Sufferist leader in Hivefled, was horrified to find that Nepeta broke up with Karkat when her kismesissitude with his best friend Sollux got in the way, as he thought they would follow in the footsteps of their respective ancestors. Nepeta, the Homestuck canon's resident Shipper On Deck, is horrified by his idea of her and Karkat being in a relationship which transcends the quadrants; not only does she not want to be left without her kismesis and moirail, but it would make her own shipping wall a lot more boring. Gamzee, meanwhile, becomes a Shipper On Deck for Feferi and Sollux's moirallegiance, fearing that he will be stuck with Feferi in a Pair the Spares situation and repeat the relationship of their ancestors. (Their shared ancestors, to be specific. Moirails reproducing is a taboo equivalent to human incestuous reproduction, but Condesce and the Grand Highblood did it anyway.)
Nepeta'sshipping wall and ways are Deconstructed in Be the Sea Dweller Lowblood, where she's oblivious to Eridan's flushed feelings for her and tries to set him up with his moirail Feferi. This eventually leads to her discovering that she doesn't understand Eridan enough to be a true moirail, while Feferi is not pleased with Nep seemingly trying to steal her moirail away, and just how manipulative and controlling this makes her appear to them.
There are multiple Shippers on Deck in Some Semblance of Meaning: Fen implies that she wishes Vale had been born in District Five so she and Fen's brother Lark could have gotten together, whereas Vale's sister Laurel is shown to be a wholehearted supporter of Vale/Obsidian (and she's not alone).
Many fanfics have the characters (Kim and Ron's parents and the kids in school) be this for Kim and Ron, even before they were an Official Couple. It was also acknowledged that some assumed that they were already a couple beforehand.
Deconstructed in Perfection Is Overrated, in which Bachiko, as well as her accomplice Meiko not only has the opinion that certain characters would be better off with other people, but as a SUE, has the power to make those desired couples a reality.
Played straight with how Isana supports Nagisa and Keitaro getting together after noting how close they've become.
Mutsumi wants nothing more than to help Naru and Keitaro hook up. Too bad this means constantly ignoring Keitaro's insistence that he doesn't want anything else to do with the violently abusive Naru. Her continued attempts to 'help' completely destroy their friendship.
Rich Bitch Kagura is a rare villainous version: she specificially wants Nagisa and Keitaro to get together so it'll be even more painful for Nagisa to lose him.
Earth and Sky: Rarity plays matchmaker for Applejack and Soarin' and Apple Bloom and Pipsqueak, and takes things to almost Chessmaster levels of planning.
The Elements of Harmony and the Savior of Worlds: Applejack and Mike actively try and play matchmaker between Fluttershy and Big Macintosh (AJ's actually been trying for years, hampered by Fluttershy's shyness and Big Mac not feeling good enough for her). Granny Smith eventually gets in on the act, complaining that the two of them have been laying it on too thick.
Big Mac appears to be this for Caramel (Big Mac, AJ and Applebloom's cousin in the fanfic) and Blues Noteworthy.
Rarity, and to a lesser extent Rainbow Dash, decides to help things along between Fluttershy and Pencil Pusher in Flash Fog.
Spike becomes this in parts of It Takes a Village, first prodding Fluttershy and Big Macintosh together. In the epilogue he's finding a new date for Rarity and also working on getting together Pinkie Pie and Caramel.
In Magnetism, Rarity apparently starts shipping Fluttershy and Rainbow Dash after hearing about their awkward bath together.
Apparently is replaced with blatantly in the last chapter when she and Pinkie try to erect a banner with Fluttershy and Rainbow Dash's face with a heart between them outside the Golden Oaks Library. Applejack, however, talks them into taking it down before either shipped pony can see it.
In The Melting Pot, a good part of the action is set off by Pinkie Pie telling Lyra and Bon-Bon they make a cute couple just before going on a trip. In the final chapter, Pinkie turns out to be subtly setting up several budding romances between background characters from the show.
It's shown during Shining Armor's flashback arc in the Pony POV Series that Cadence's hoofmaidens and at least some of Shining's squadmates shipped the two of them prior to their getting together. And they weren't subtle about it. At all.
Shining and Cadence themselves both get their own, more mild, examples of this — Cadence is working with the hoofmaidens to set up Private Running Gag and Private Garnet Chambers, while Shining has said on a couple of occasions that Captive Audience is the stallion he feels would be the best match for Twilight.
Taken up to eleven in with Rarity in the fanfic Shipping Goggles. Rarity apparently has a pair of the eponymous goggles welded to her head, as even the most innocent of interactions is treated as prime shipping fuel by the white unicorn, much to the frustration of Twilight, who can see Rarity shipping their friends.
In the Ponies Of Olympus series, as revealed in Atlas Strongest Tournament, Sweetie Belle ships Spike with her sister, while Apple Bloom ships him with Scootaloo.
The Nuptialverse: A Running Gag in Families is that Pinkie Pie is convinced that Applejack and Twilight are in denial about their true feelings for each other.
In Naruto Veangance Revelaitons, an odd variant of this happens when Ronan's girlfriend Sakura, meets Atni for the first time (While Sakura was dead after being killed by the Council, Ronan met Atni and had sex with her several times).
Sakura:OMG WHO IS DIS WHOT BITCH CAN U FUK HER/"
Tsunade and Hanabi ship Naru Hina in A Growing Affection. Tsunade offers to train Naruto and Hinata together, partly to help them get stronger, but also to give them a reason to spend time together. Hanabi is more than willing to run interference for Hinata with their family, at least until she gets angry at Naruto for an unrelated incident.
In Nobody Dies, Rei was apparently shipping Shinji/Asuka ten years before the three of them even met each other. She also somehow hooked up Toji and Hikari during the unseen Third Angel fight, and now claims to be shipping Shinji/Asuka's and Toji/Hikari's hypothetical future children. This seems to be a universal trait among the Ree (they all ship Shinji/Asuka, except Zyuu, who wants Shinji for herself), and Lilith herself seems to ship everyone/everyone as part of her desire for babies.
Most of the humour in the fic Written in the Stars (in which the Kirks of both realities have been genderswapped) comes from Kirk Prime trying to get her counterpart to hook up with Spock. This is justified in that Kirk Prime and Spock Prime were married in their own reality, and Kirk Prime just wants her counterpart to be happy. She succeeds.
In Things We Donít Tell Humans, Optimus Prime is this for Bumblebee and Barricade, and perhaps for Sam and Carly, too. Jazz and Prowl have a whole fleet of shippers, consisting of their son, Jazzí parents, Ratchet, Optimus and Elita, Megatron, Terratron, and likely more. Prowl, in turn, ships Ratchet and Wheeljack.
The Gaang Jr. fan-project gives us Yi Lin, a walking riff on the Avatar fandom's shipping community. She's obsessed with other people's love lives and hides all her collected "Proof" in a book supposedly about...well, ships. She's a Tyzula fangirl in particular, although she can't get other people to see it (and if they ever do see it, they're not going to tell her since it would just make her harder to live with). Fittingly, she refuses to see that there's something going on between Irah and Hayoda.
In the Axis Powers Hetalia/Ace Attorney crossover Turnabout Hero, Hungary seems interested in shipping Edgeworth and Phoenix. At one point, she even snaps a picture of them sitting together and thinks it's cute. Given that it's Hungary, no ones surprised.