There are so many moving moments in U2's live performance history. To me the most iconic moment was the leap at Live Aid. I can't remember anthing like that happening before that concert. That was an OMG moment. It really showed Bono's humanity for that poor girl that was being crushed.
So, some other moments that have left an impression on me:
This is an audio of One from the Tacoma show from April 21, 1992. This one to me is very emotional. I love the high F# Bono hits in falsetto voice towards the end. I haven't found a live video of this performance yet.
U2 - One (Live Tacoma 92 GREAT VERSION!!) - YouTube
The next is With or Without from Sao Paulo 1998. This video wrenches at my heart. The whole exchange between Bono and the fan is intense. He seems so broken up at around 1:30 that he can't get the words out. He also seems to be crying.....
(or he's sweating). In either case it's very moving.
U2 - With Or Without You - Popmart Live From Sao Paulo 1998 - YouTube
I think that version of Dirty Day posted above is one of my new faves. I've never seen it before.
What a great band.