How about you starting the list you are asking for to start the discussion?
I'll start a list that is very likely to be incomplete consisting of only the cases that come to my mind spontaneously and that won't cover the last few movies which I know much less than the earlier ones:
Original movie:
Littlefoot cries after the death of his mother.
Littlefoot cries after the fight with Cera.
Cera cries after her rescue by the others.
Ducky cries after the fight with Sharptooth assuming Petrie to be dead.
2nd movie:
Ducky cries after thinking that the egg from her family's nest was destroyed.
Chomper cries right after his birth when everyone else ran away and also after the incident in which he bit Cera (Littlefoot was crying there too I think).
Littlefoot cried during the goodbye scene with Chomper.
3rd movie:
Cera cried during the nightly talk with her father.
4th movie:
Both Littlefoot and his grandmother cried during when they believed grandpa was doomed to die.
5th movie:
While there were plenty of scenes with very sad characters I am not sure if there were any where they reached the point at which tears were rolling.
6th movie:
The unnamed longneck kid in the flashback of the lone dinosaur was crying.
Dinah and Dana were crying when admonished by Cera.
Littelfoot was crying during the goodbye from Doc.
7th movie:
Petrie cried after his friends caught him "spying" for Pterano and quarreling about it and again before the song "good inside".
8th movie:
Ducky crying in the scene of the song "family".
9th movie:
Littlefoot crying in the scene in which they all believed that Mo had been killed by the swimming sharptooth.
10th movie:
Littlefoot crying after learning about Bron being his father (personally I feel that Bron's pathetic excuses ignoring everything established in the original movie would have made a better reason to cry).
Shorty crying when setting out to leave Bron.
I believe more than one character cried during the song "Bestest Friends" but I am not quite sure on that one.
