Red Is the Rose That in Yonder Garden Grows
Moscow Nights
Believe Me, If all Those Endearing Young Charms
Kisses Sweeter than Wine
Two Sleepy People
basically any waltz

folk?
Give Yourself to Love (Kate Wolf)
If We Only Have Love (Jacques Brel, trans. Shuman & Blau)
Somewhere (West Side Story)
Mistress Mine, Where are You Roaming
(what is love? 'Tis not hereafter,
Present mirth hath present laughter
So come and kiss me sweet and twenty,
Life's a stuff 'twill not endure.)
(in Shakespeare somewhere, possibly Twelfth Night)
a half-remembered song from summer camp
("We've shared this time together
A joy-and-grief time, a falling-leaf time.
Though you may doubt me, I love you.
You'll live without me, but I love you.
Much more than we ever could have guessed...")

songs which are personally romantic due to context first heard in:
The Sun (They Might be Giants)
some awful trite pop thing about "sweet dreams of taking over/ dreams of dancing through the night."
Mahler's 8th symphony
Closer (Nine Inch Nails)
Red is the Rose goes in this category too
Actually, this could branch into an interesting thread on its own: Life's Soundtracks, or, What was playing when ...

Chocolate Pi (feeling all warm and gooshy inside)