Leela and Andred
In The Invasion of Time, the Doctor brought Leela home with him to Gallifrey, where he perpetrated an insanely over-complicated plan to foil the invasion of the Time Lords' planet by the Vardans. Assisting the Doctor was Andred, Commander of the Chancellory Guard. Leela spends most of the story in the outlands of Gallifrey rounding up help, while Anded spends most of the story with the Doctor, looking confused. They have no on-screen time alone together, and there doesn't seem to be any time for them to sneak in any off-screen action. And yet, at the end of the episode, Leela declares that she's fallen in love with Andred and will be remaining behind on Gallifrey. Actress Louise Jameson has stated numerous times that she was unhappy with how Leela was written out, and indeed, she plays this final scene as if Leela is mildly embarrassed about the whole situation. Big Finish Productions' audio adventures have continued the story of Leela's life in their Gallifrey series, and they wisely sideline the whole relationship right off the bat.
I've now decided that "to leela" is a verb meaning to suddenly write out a regular character on a TV show by creating an out-of-character romantic entanglement for them with a character with whom they have no previous history or significant interaction. Please spread it around.
Peri and King Yrcanos
Peri's travels with the Doctor ended in the season-long serial The Trial of a Time Lord. She was about to be killed by having the brain of a dying amphibious alien transplanted into her body; before the Doctor could intervene, he was kidnapped by the Time Lords to be put on trial for interfering in the history of other races (the Time Lords have a thing about that). Peri is apparently killed, but at the end of the trial it is revealed that, in fact, she survived and, since the Doctor apparently never bothered to go back for her, she wound up marrying their ally King Yrcanos. Peri's departure seems totally out of character, and there was no suggestion of any romance brewing between the two of them during the story. Peri was totally leela-ed by the writers. Still, Yrcanos was played by Brian Blessed, and while he doesn't look as good in this story as he did as prince of the Hawkmen in Flash Gordon, he's still not too shabby of a catch.
Vicki and Troilus
There's nothing inherently wrong about Vicki's love for Trojan Troilus; her decision to leave her life in the TARDIS behind to be with him forever as Troy falls around them is actually quite romantic. It's the fact that she takes the name Cressida that suggests that their romance may not meet the happiest of endings.
Fitz and Trix
If you're like the majority of modern Doctor Who fans, you're probably asking yourself, "Who the hell are Fitz and Trix?" Fitz Kreiner and Trix MacMillan were companions of Paul McGann's Eighth Doctor in a series of tie-in novels published by BBC Books between the airing of the 1996 TV movie and the start of the new series in 2005. Fitz traveled with the Doctor for a very long time, and the two became the best of friends. Trix was a con artist who they met while she was working for Sabbath, an enemy of the Doctor's. She left Sabbath's employ and asked to travel with the Doctor - he refused, so she stowed away. She was eventually discovered and accepted (as most stowaways on the TARDIS are), and she and Fitz eventually fell in love. In the final book in the series, The Gallifrey Chronicles, Fitz and Trix resolved to marry and stay behind on Earth as soon as the current adversary was dealt with. Nothing wrong there. It's just that Fitz is one of my favorite companions, and I have a weird fictional character crush on him. There, I said it. Ok, it's irrational and perhaps a little creepy, but I stand behind it. Fitz was a fully-realized character and was awesome, and Trix was under-developed and sucked. There are no pictures of Trix for me to share with you, since she only appeared in the books (and she probably looks like a total whore), but Fitz made one appearance in a Big Finish audio, The Company of Friends, in which he was played by Matt di Angelo. It's a small picture, since he only appears in one little sliver of the CD cover, but isn't he handsome? Maybe we should move on...
Martha and Mickey
Although I haven't been ranking this list, it's safe to say that this is my least favorite Doctor Who romance, even though I really love both of these characters (yes, Mickey too). It's the pure randomness of it that gets to me. Martha and Mickey meet in Journey's End, where they have no interaction whatsoever outside of the big group scenes. The next time we see them, in The End of Time, they're married and happily hunting Sontarans together. Granted, when we saw them at the end of Journey's End they were both walking away with Captain Jack, so who knows what kind of unspeakable depravity might have followed. Still, at the time, Martha was engaged to another man - to Tom Milligan, whom she met (sort of - it's complicated) in Last of the Time Lords. Russell T Davies has said that he married Martha and Mickey because his last name is Smith, and hers is Jones, and Martha's first episode was Smith and Jones, so he thought it would be funny. Ugh. Having read a bit about Davies' writing process, I doubt this was the real reason, but still. It felt to me more like he wanted to make sure Mickey got a cameo like all the other past Tennant companions at the end of The End of Time, but didn't want to spend time on a whole scene just for him so shoehorned him into Martha's. I'm not sure which explanation is worse. For whatever reason, Davies managed to leela two companions at once!
There's also a slight ick factor to their relationship, in that Martha loved the Doctor, and Mickey loved Rose, but the Doctor and Rose were in love with each other. Martha's love for Tom seemed to fulfill the promise of the clean break she was looking for at the end of Last of the Time Lords, but her marriage to Mickey smacks of sublimation on the part of them both. I see marriage counseling in their future.
So happy Valentine's Day to you and yours! Please choose your partner more wisely than most of the Doctor's companions have, and here's hoping you don't get leela-ed!