I love Out Of Africa. In fact I love John Barry in general and I cry at his Wednesdays Child. So moving!

I just remembered a song by the Four Tops that has me crying too. A Simple Game. A real weepie for me.

I've been feeling in that sort of melancholy mood for music

Sound of Silence - Simon and Garfunkel

Fields of Barley - Eva Cassidy

THIS version of Imagine - John Lennon

Dance with my Father - Luthor Vandross

Nothing Compares 2 U - Sinnead O'Connor

It's the first time I hear the original version, and I think it is at least as good as Bette's version.
A few more that I think qualify as weepies and haven't been mentioned -
Solitaire - The Carpenters (not Andy's version!)
Empty Chairs At Empty Table - either Michael Ball or Colm Wilkinson's version will do
Bring Him Home
Aria of Liu from Turandot - don't understand a word of it but so heartbreakingly sad https://www.youtube.com/watch?v=a_Z5WrPt37w

Bring Him Home is a definite weepy for me too, especially sung by Colm Wilkinson.
I'm a bit of a philistine when it comes to opera, I only know what I heard on Inspector Morse, but this is also beautiful (Puccini again): https://www.youtube.com/watch?v=DUe1RB24qU0
This Woman's Work by Kate Bush can make me weep, depending on my mood. And the video of Kate's Cloudbusting makes me cry too, but that's down to the combination of the song and the storytelling in the video.

None, because I'm a man.

I'm scrapping the bottom of the barrel now for sad songs and come up with

The West Wind Circus https://www.youtube.com/watch?v=t7BIebGfFH4

and even though Copacabana has an upbeat melody, it tells a sad story.

Ooh, I can't believe I forgot these two:

I Dreamed a Dream - Ruthie Henshall

And Rose's Theme from Doctor Who which really really makes me cry, right from the first few notes, and when the strings go soaring away at about 0.58 I usually end up a complete sobbing wreck.
https://www.youtube.com/watch?v=bPWDUMzJ1i4

Here Without You - 3 Doors Down

Autumn Leaves https://www.youtube.com/watch?v=OkQaFJjhsek

Yours - Dionne Warwick https://www.youtube.com/watch?v=e1qi06d3ogE

Sarah Brightman and Andrea Bocelli - "Time to say Goodbye" does make me emotional.

Another sad song from Sarah -

Wishing You Were Somehow Here Again https://www.youtube.com/watch?v=7Z8bBQmbXRk

love that song even I like Paul Danos Version is just gave me feels the first time I heard it in the cinema and so does Don't Worry Baby

I remember we had a remembrance assembly for the former student died in the war,they were playing One Direction-Little Things for these assemblies for remembrance they had reflection music and during college and we had a remembrance as we walked into Say Something listen to the song just too much

other songs were
A Single Man Tear from 200th Supernatural episode(When my Grandma died back in 2012 my dad was crying all the way home and these Lyrics had these meanings sung by 2 girls playing the Winchester brothers singing about their father)

Ill Just Wait Here Then(made me emotional the first time Ive headed it)

On the last day before summer last year,i was sad and when my form tutor left for the video I chosen its been a long day without you my friend,ill tell you all about when I see you again from FaF7

Just 3 David Bowie songs I been listen on the day he passed away on the news(Life on Mars. Lazarus and Heroes)gonna say Lazarus was upsetting because of these lyrics(Look Up Here I'm in Heaven,i got scars that cant be seen) but the video was hunting and make my heart ache at the very last bit when he enters the closet
its just like hes Saying Goodbye but on my birthday I brought blackstar from iTunes and yes more clues of his death(Dollar Days and I Cant give everything away)

Time in a Bottle - Jim Croche

https://www.youtube.com/watch?v=WIF4_Sm-rgQ

Always gets me, it's also my mum's favorite.

As for the classical piece, Lascia ch'io pianga.