There are quite a few that make me well up and sniffly and tearful. I find non-real life makes me cry, sniffle, well up far more than reality does. Thus, it was a tough choice.
I'm going to go with a scene that can actually make me start to sniff and make my eyes fill with tears every time I see/think of the scene. It's all the more effective in making me cry, because it's set around a song.
It's from the Buffy episode One More With Feeling
We know all along how deeply Giles loves Buffy; how he sees her as his daughter - he even lost his job as Watcher because of it. So his realisation that he can no longer be in her life as her father, can no longer slay the demons for her, that because of how much he loves her he has to walk away, is so devastatingly painful to see and hear. And if that wasn't enough, the song combines with Tara's discovery that Willow had literally put her under a spell and how their songs mirror and combine with the soul destroying, heart breaking line 'And it'll grieve me, cuz I love you so' as they are staring at Buffy & Willow who are chatting. I can't watch the scene/hear the song without being heart broken.
And it's all the more devastating because ASH & AB are such wonderful singers.