You know what, I'm glad. My first thought was that I liked the classics, but now that I think about it further, I don't much care for many of the accepted great couples in movies. This might be weird, but here goes.
Favorite On Screen Couples
20) Lady and Tramp - Lady and the Tramp (1955) - Also, best on screen kiss.
19) Henry Roth and Lucy Whitmore - 50 First Dates (2004) Of all the Barrymore lovable retard girlfriend roles, this one tugs at me the most.
18) Adam and Katherine - 50/50 (2011) This onscreen romance comes from such an interesting place. I love Kendrick's portrayal of a counsellor in over her head.
17) Tom and Violet - The Five Year Engagement (2012) Segal is like Cusack. Loveable always. Emily Blunt is a fine, emotional actress. Movie good.
16) Bud and Lindsey Brigman - The Abyss (1989) One of the most touching scenes ever, when he's down in the water, communicating with...hold on. One sec. It rained in my eye.
15) Hawkeye and Cora - Last of the Mohicans (1992) My wife would divorce me if this wasn't on here.
14) Mickey Ward and Charlene Fleming - The Fighter (2010) This might be mostly to the credit of Amy Adams, but I believed this couple. Nah that's mean. Wahlberg was great too, otherwise, I wouldn't have wanted them to be together so badly.
13) Harry and Sally - When Harry Met Sally (1989) A classic. Delightful and memorable, charming and witty and sweet.
12) Noah and Allie - The Notebook (2004) When I finally saw this movie, I was actually kind of pissed how much I cared about these two.
11) Willis Davidge and Jeriba Shigan - Enemy Mine (1985) More of a bro-up than a couple, but still, it gives me all the proper, lovey feelings, and all the complexity of a true romance.
10) Westly and Buttercup - The Princess Bride (1987) Not even the dread Pirate Roberts could keep them apart.
9) Micky and Mallory Knox - Natural Born Killers (1994) My favorite of the murderous duos, these two wrecked some shit in the name of love.
8) Ennis Del Mar and Jack Twist - Brokeback Mountain (2005) One of the best love stories ever, one of the great on screen couples. Ang Lee makes you believe cowboys can be happy, if only briefly.
7) Andie and Blane - Pretty in Pink (1986) I'm a huge sap for John Hughes. Who isn't? This is the definitive on screen romance from his movies, though there were a lot to choose from.
6) Martin Q. Blanke and Debi Newberry - Grosse Pointe Blank (1997) Another of my favorite all-time films, I absolutely love Minnie Driver in this movie. And Cusack is Cusack, loveable as ever.
5) Phil Connors and Rita - Groundhog Day (1993) This is one of my favorite movies. The love story is central, and when Phil finally gets genuine, the romance is tangible. Bill Murray rules.
4) Joel and Clementine - Eternal Sunshine of the Spotless Mind (2004) - It took me awhile to come around on this one, but now I find it a more than accurate fable of love lost and perseverance in the face of hopelessness. Jim Carrey's best role anyone?
3) Will Hunting and Skylar - Good Will Hunting (1997) - Minnie Driver, you are my Hugh Grant. I love this movie, love this couple, and it's intrinsic to the success of a film that may not even be considered a romance.
2) Pat and Tiffany - Silver Linings Playbook (2012) This felt so real. Easily the best movie of the year for me, all because of these two fuck-ups finding each other in the mire.
1) Melvin and Carol - As Good As It Gets (1997) Yea, it doesn't get much better than this. He's nuts, she's broken. Somehow they fit together, like the cracks on the sidewalk.
Short Answer: Barrymore, Cusack and Segal would've owned this list if I'd allowed for doubles.
(Hey, Minnie Driver is on there twice!)
Note: There are no examples from 'Classic' movies. That's because love wasn't invented on celluloid until the late sixties. Except for dogs.