The most obvious is probably "Don't Cry For Me, Argentina". It's the first moment in the show where Eva is completely honest in public and evokes warm fuzzies from pretty much everyone.
YMMV. I (along with many others, probably) am fond of the interpretation that "Don't Cry for Me, Argentina" is the OPPOSITE of honesty: it's a calculated political speech designed to appeal to the masses, without ACTUALLY saying a whole lot.
Peron's love for Eva, full stop. He comes so far from their mutually-beneficial-liaison in "I'd Be Surprisingly Good for You" to his heartbreaking devotion to her when she falls ill. It takes a heart of stone not to "awwwww" when Peron carries Eva up the stairs of their home during "You Must Love Me" in the film.
Another film moment: Che dancing in the rain with the little girl during "And the Money Kept Rolling In and Out."
At one point, he helps a girl who is injured during a political rally.
The exchange between Peron and Che in the finale ("she had her moments, she had some style"), the two men who knew and loved Eva best.
Che bending over Eva's coffin and kissing it at the end of the film.
Also from the film, Eva's anguished protests when Peron is arrested during "A New Argentina" and her rallying the descamisados to free him, showing that she genuinely cares for him even though she's using him for political gain.