How about "Summer of '42"? I haven't seen this film in a number of years, but the impact of it's ending still remains with me. A coming of age story set on Nantucket in 1942 depicting the loss of innocence faced by both the teenage boy in the story, who has a crush on a young woman who's husband is off at war, and the world in general. Jennifer O'Neill was completely lovely in this film.
To some of the solid titles already mentioned in this thread I'll add another selection. I am uncertain as to whether I'd classify this film as tragic, but it certainly is downbeat. Personally, I find this film to be overwhelmingly poignant and emotionally affecting:
Some of the following are cheesy, but they did make boo hoo something fierce:
Romeo and Juliet (60's version) A Walk to Remember Love Story (70's version) Cider House Rules Legends of the Fall Queen Margot Margret's Museum Titanic
Who doesn't feel Gregory Peck's pain as he takes that long walk away from the woman he's just fallen in love with? Up until the last 15 minutes of the film its your typical 1950s romantic comedy, then BAM!!!
Though not a movie, The Star Trek: The Next Generation episode "Perfect Mate," which featured Famke Jansen and Patrick Stewart's first work together, also stands out as being particularly bittersweet.
Jesse and Celine's relationship unfolds so freakin realistically and feels so genuine, the very fact that they have to part ways at the end (i'd drop everything in my life back home to stay with such a gorgeous creature as Celine, there's no way i'd leave her)
makes it tragic.
The worst kind, longing for someone you just can't be with.
I second the obvious statement that Casablanca is the all-time classic example. A Place In The Sun is probably the all-time most depressing romantic tragedy though.
Brian's Song is a good platonic twist on the theme -- it's essentially a love story about two best friends.
The anime film Millennium Actress is another classic twist on the theme. It actually revolves around two romantic obsessions. (And if you are into Japanese cinema, it is a MUST SEE.)