I cried a lot watching Buffy, and still cry every time in some spots (end of Becoming, The Body, etc.).
For me, the saddest moment of the entire series is one that I may or may not tear up about - the beginning of The Gift. I love how in the craziness of the season, especially what has just happened (Buffy's terror when Glory sees Dawn as the key and her subsequent decision to run, her coma, etc.), we get brought back to just Buffy vs. a vampire in an alley. She goes through the witty banter, but her heart is clearly not in it. Then, the guy she's just saved looks incredulous and says she's just a girl, and she replies "That's what I keep saying."
I find this scene absolutely crushing and heartbreaking. The weight of that scene (and what it says), for me, sums up the entire series. On initial viewing, with the "Last time on Buffy..." montage that started with episode 1 and went through every episode, it was especially painful to watch. I'm going to tear up just thinking about it!