Jem Finch's sensitivity is evident in several of his interactions with characters in the novel.  The most obvious of these occurs after the trial when the verdict has been delivered.  He's visibly upset, and has this conversation with his father in chapter 20, 

"Atticus-" said Jem bleakly.

He turned in the doorway. "What, son?"

"How could they do it, how could they?"

"I don't know, but they did it. They've done it before and they did it tonight and they'll do it again and when they do it—seems that only children weep. Good night."

An example that isn't as obvious is when he invites Walter Cunningham, Jr. home for lunch in chapter 3.  He knows that Walter is hungry and won't accept charity, and he's sensitive enough to invite him over as a friend.  This way he can still feed a hungry child without shaming him. He makes sure to include in the invitation that their daddies are friends. 

"Jem suddenly grinned at him. 'Come on home to dinner with us, Walter,' he said. 'We'd be glad to have you.'

Walter's face brightened, then darkened.

Jem said, 'Our daddy's a friend of your daddy's. Scout here, she's crazy – she won't fight you any more.'"

Scout and Jem are fascinated with Boo Radley throughout the novel, and feel a strange connection with him.  Boo obviously feels the same way, as shown when he starts leaving gifts for them in the knothole of a tree on their walk to school.  The very personal gifts, like the spelling medal and the figures carved from soap, indicate to Jem that Boo is not just some phantom entity, but a real person.  For these reasons, Jem actually cries when they arrive one day to find that the hole has been cemented closed by Boo's older brother in chapter 7.  This is further evidence of Jem's sensitivity.  After all, not many young boys would cry for such a thing. 

Next morning on the way to school he ran ahead of me and stopped at the tree. Jem was facing me when he looked up, and I saw him go stark white.

"Scout!"

I ran to him.

Someone had filled our knot-hole with cement.

"Don't you cry, now, Scout... don't cry now, don't you worry-" he muttered at me all the way to school.