Tom Felton and Emma Watson, who play Draco Malfoy and Hermione Granger respectively, have a close friendship, even after Emma admitting her "terrible crush" on Tom when she was twelve. Fans still speculate there is something between them even though Tom is happily involved with his girlfriend.
Despite their banter on-screen as Damon Salvatore and Bonnie Bennett, Ian Somerhalder and Kat Graham display a close frienship in real life. Despite the fact each have their own relationships fans still seem to ship them.
There is a staggeringly large demographic that ships Thomas Jefferson/John Adams, despite them having both been married and Jefferson's long-term relationship with his dead wife's enslaved half-sister, not to mention the decades-long hate-on where they never saw each other (though that may only have encouraged said demographic) plus Adams and his wife having written some of the spiciest, raciest love letters ever to come out of American politics to each other. This example probably more properly belongs under theater as it's likely an outgrowth of 1776 the musical.
As per the introduction, Alice and Bob. Probably because canon keeps using them to illustrate concepts instead of mining for the obvious romantic potential.
Although they've been broken up for several years and have moved on to other relationships, Miley Cyrus and Nick Jonas still have a large fanbase with a large of percentage of the Hannah Montana stories on Fanfiction.net consisting of this pairing (despite it being against site rules)
Similarly Demi Lovato and Selena Gomez have a massive fandom.