If you don't think it's a jinx, I love "The Next Ten Minutes" for a wedding.
Otherwise, "Til There was You" from The Music Man is gorgeous and would also be appreciated by non-theatre people.
I always think though that first dance songs should have sort of emotional resonance with the couple. Is there a musical that you and your intended both really love?
I'll have to check out "Always."
Jimmy, what are you doing here in the middle of the night? It's almost 9 PM!