In the film, after Clarice has finished telling Lecter about her trauma as a child and admitting that she hopes saving Catherine will help her move past it, Hannibal can only say, "Thank you, Clarice. Thank you." There's an odd sense that Lecter is genuinely moved by Clarice's admission, and there's even a hint of tears in his eyes.
Despite being undeniably horrible, creepy, and cannibalistic Hannibal Lecter is capable of providing a heartwarming moment of two (though a sense of creepiness is almost always guaranteed). The ending of Clarice and Hannibal's last meeting in The Silence of the Lambs has a heartwarming edge too because the two seem to have come a sort of understanding.
"Then why not finish the arch? Take your case file with you, Clarice, I won't need it anymore." He held it at arm's length through the bars, his forefinger along the spine. She reached across the barrier and took it. For an instant the tip of her forefinger touched Dr. Lecter's. The touch crackled in his eyes.
"Thank you, Clarice."
"Thank you, Doctor Lecter."
And that is how he remained in Starling's mind. Caught in the instant when he did not mock. Standing in his white cell, arched like a dancer, his hands clasped in front of him and his head slightly to the side.
What adds to the moment is that Clarice has been warned multiple times throughout the book not to come anywhere within the vicinity of Lecter; yet she puts her trust in Lecter and takes the casefile directly from him. Hannibal doesn't even try anything with her, just watches her without any judgment as she leaves for the last time. Keep in mind that this was moments after Clarice had been coerced into revealing one of her worst memories to Lecter so she had every right to be pissed at him.
It's also worth mentioning in the film that when Clarice takes the case file from Lecter, he gently caresses her finger with his own finger. It's the only time they ever touch each other in the film, and it's electrifying. Lecter, whom we've been told is a heartless monster who'll attack anyone on a whim, touches Clarice so gently that it's like the touch of a lover.
Hannibal has another heartwarming moment when he says goodbye to Barney in The Silence of the Lambs. What makes it heartwarming is how sincere Hannibal is being. Hannibal respects Barney because Barney treated him with dignity and professionalism. Hannibal returns the courtesy.
Barney: Yes, Doctor Lecter?
Hannibal: You've been decent to me for a long time. Thank you.
Barney: You're welcome.
Hannibal: Next time Sammie's at himself, would you say goodbye for me?
Hannibal: Good-bye, Barney.
The footnote of the book mentions that he sent him a generous tip, as well as yet another "thank you". The sequel has Barney mentioning that he was never the least bit worried that Lecter would come after him.
A small similar moment in the film. During the scene in which Chilton reveals to Hannibal that the offer from Clarice was fake and taunts him for believing her, Barney is standing next to Hannibal looking genuinely angry at how Chilton is treating him.
Hannibal himself has a moment both there and later on when Clarice visits him for the last time. When Chilton taunts Hannibal over it, he doesn't get mad or even heartbroken, but genuinely impressed at how Starling was able to lie to him. Later on in his new prison, he compliments Starling on it.
Hannibal: Anthrax island, that was a nice touch, yours?"
Barney in general is just a genuinely Nice Guy. In the film when Clarice first goes to speak with Hannibal, he seems to detect her uncertainty, reminding her that he'll be just down the hall. He even reassures her that she'll do fine while flashing her a friendly smile.
The relationship between Starling and Crawford has some heartwarming moments as well. Starling looks up to Crawford and Crawford makes sure to treat her with respect and listens to her ideas when many would dismiss her. One moment that stands out though is when Starling, after talking with Crawford about how others will take his cue on how to treat her so his actions matter, watches Crawford walk away after he acknowledges her concerns but then tells her to get back to work in The Silence of the Lambs.
She watched him walk away, a middle-aged man laden with cases and rumpled from flying, his cuffs muddy from the riverbank, going home to what he did at home.
She would have killed for him then. That was one of Crawford's great talents.
Doubles as a tearjerker, but everything that Crawford does for Bella in The Silence of the Lambs perfectly shows how much he loves her without putting the thought into so many words in the book. Crawford makes sure Bella has the best nurses and doctors at her care, he makes sure she can come home to die instead of in a hospital, buys special accommodations for their house so she can be properly treated there, and learns how to give her shots. Heck, the list of things he does for her goes on and on. Not once in the book does he ever indicate that resents having to take care of her. He just does his best to make her comfortable and pours all of his love into his actions.
It's well-known that Anthony Hopkins went through great improvisational lengths to freak out Jodie Foster during their time together while shooting. However, you can see what true admiration she has for him by her beaming smile when Hopkins wins the Academy Award for Best Actor. While the names of the nominees were being read, Hopkins was the only one who got both cheering and applause from the audience as opposed to the usual polite applause, and when he was announced as the winner, the audience gave him a standing ovation, a sign of who the Hollywood ensemble thought undoubtedly deserved the statuette that night.
Lecter calling Clarice and assuring her that he's not going to come after her—"The world's more interesting with you in it."
Early in the book as Jack Crawford's wife is dying, Hannibal Lecter of all people sends Crawford a note containing part of a John Donne poem and his sympathies. It's clear that Hannibal is capable of expressing a more human side to those he respects.
"O wrangling schools, that search what fire
Shall burn this world, had none the wit
Unto this knowledge to aspire,
That this her fever might be it?"
I'm so sorry about Bella, Jack.
However, this could be also seen as Lecter simply taunting Crawford, particularly since several chapters later, while talking to Clarice, Lecter viciously comments that he would gladly see how much of a stoic Crawford really is once Bella kicks the bucket.
One of the firemen who respond to Clarice's distress call is told to take Gumb's dog to the pound, but instead takes it home with him, a small moment of humanity after the terrifying climax.
In the film adaptation, it's implied that Catherine is the one who adopts Precious. When she's rescued, she's seen carrying the little dog out of the house.
Clarice's graduating ceremony, including a nice Call Back to the entomologist hitting on her: he and his lab partner came to the ceremony.
The very ending line, wherein Clarice, secure and dating the entomologist, is sleeping happily "in the silence of the lambs."
In a meta example, the fact that the film, which featured dozens of caterpillars, moths, a cat, and a dog in its production, is proudly and truthfully a film that harmed none of the animals in its production. All of the animals, even the caterpillars, were treated exceptionally well. The bugs were even given honey to eat.