FFI - No romance.
FFII - No romance.
FFIII - Random hookup between Onion 4/Ingus and Sara. Completely unnecessary and a bit confusing. Had it been developed it would have been better received, but as is it's just random.
FFIV - Cecil and Rosa, pretty significant plotline. Could have done without it, but it makes for a solid love triangle.
FFV - No romance. I think Bartz gets a hard on for Faris in a dress, but that's really all it is.
FFVI - Celes and Locke. More of a side thing. Not necessary, but not unnecessary either.
FFVII - No explicit romance. Schoolgirl crushes and schoolboy confusion. Girl knows boy. Girl likes boy. Boy likes other girl. Other girl likes boy but doesn't like like boy. Other girl likes other boy. Other boy is dead, and boy has other boy's memories. Other girl dies. Boy remembers other boy. Boy doesn't like other girl. Boy is sad. Girl still likes boy. Boy is still sad. To be honest, this game could have done without the romance. While it makes for some interesting character development, it's just kind of confusing as a whole.
FFVIII - Squall and Rinoa. The very basis of the game. Without these affair, we'd probably have an extremely dull cast. Squinoa allows for some much needed character development and to be honest, Squall sort of ends up Rinoa's 'knight' in a sense, which is nicely integrated into the story of the sorceresses.
FFIX - Zidane and Garnet. An explicit romance, but fairly sidelined. It isn't particularly necessary, but it's well received and doesn't distract from the plot.
FFX - Tidus and Yuna. A fairly large portion of the plot. While the game is about Yuna's pilgrimage, Tidus' origins and Yevon's blasphemy, the game is also essentially a love story, much like FFVIII.
FFXI - P2P hookups?
FFXII - Ashe and Rassler. A romance that may or may not have been arranged, but nevertheless happened almost entirely before the game. His death is brought up on occassion but not entirely a signficant part of the plot. It's well placed and doesn't get in the way, as Rassler serves as a sort of... 'progression' pusher for Ashe.
FFXIII - Snow and Serah. The most annoying relationship I've ever seen. While it serves good inspiration for Snow to fulfill his Focus, it's super annoying and causes way too many problems for the rest of the party, as well as himself. And ultimately, becomes a complete moot point as the situation becomes so much bigger than what they originally perceived it to be.
All in all, most of the relationships work. The ones that don't work could have been either further developed or scrapped entirely, but none of them particularly 'distract' from the main objective in my opinion.