Moments pages are Spoilers Off. Proceed at your own risk.
Henry's determination to save his father.
Young Henry: I want you to come home.
Even though Jack tends to pick on Henry, he obviously takes a liking to the boy, giving him advice and making sure to keep him safe. Most likely because he's Will and Elizabeth's son.
Also he stuck around at the end long enough to see Will freed and reunited with Henry and Elizabeth, even if he mocked them immediately after. He really does care about his old friends, although he'd never admit it.
Jack's uncle, during their brief reunion, warns his nephew there's a blood moon out, and it would be safe if he remained on land.
Much of Jack's iconic outfit - his headwear and beads in particular - were tributes from the crew of his first ship after he saved them all and became captain.
After the events of the movie, Jack seemingly no longer asks tribute from his crew, now that he's back to his former glory of sailing free upon the Black Pearl.
The compass returns to Jack, as though forgiving him for having 'betrayed' it in the beginning.
Barbossa seems to have become a more merciful captain since the previous films (and a much more Benevolent Boss the Queen Anne's Revenge had compared to Blackbeard) as he's been sharing the treasures from his ten ship fleet with his crew, with the ship and crewman all decked out.
While being chased by Salazar's sharks, Henry panics that they may die, as he and Jack desperately try and keep their rowboat from sinking. Jack assures Henry "No, we'll be fine, we'll be fine." It may be a white lie, but at least it surprisingly comforting coming from the pirate who's usually out to save his own skin.
Heck, back when the sharks first showed up and nearly ate Henry, Jack scrambled to save him. And did his best to keep Henry alive throughout the whole escapade.
Although it borders on Nightmare Fuel, Salazar and his crew being freed from the Devil's Triangle counts as this to an extent. They are nothing short of happy to have sunlight on their faces again, even if they are still cursed.
And later, we have the same scenario in the form of Salazar and his crew rejoicing their curse being lifted after the Trident is broken.
The scene where the island that holds the key to the Trident turns out to hold diamonds and gemstones that make it resemble a night sky on earth. But what makes it heartwarming is how Carina lamp shades how beautiful it is.
Barbossa left Carina at an orphanage after her mother died because he wanted her to have a better future than being the daughter of a pirate. Even after he finds her again, he doesn't want her to know that he's the father she's been chasing for so long because he's afraid he won't live up to the image of him she has in her head. He finally admits it right before sacrificing himself to save her. At the end, Carina adopts his last name in his memory.
Carina (as she recognises the tattoo on Barbossa's arm and realizes who he really is): Who am I to you? Barbossa: Treasure.
Looking back, Carina seeing her father one more time was all thanks to Tia Dalma reviving him in the 2nd movie.
Actually, since Carina is younger than Henry, she was conceived after the events of At World’s End, ergo, if Tia Dalma hadn’t resurrected Barbossa, Carina wouldn’t even exist.
In The Stinger, Elizabeth is smiling while she's in bed with Will. Either she's happy to have him home... or the sex was very good. (Or both.)
Jack the Monkey sitting on Jack the Pirate's shoulder. While pirate Jack has stated multiple times to hating the monkey, shooting it whenever he gets the chance to and threatening to eat it, he's willing to let the animal perch on him even if he doesn't seem entirely comfortable with it.
Jack knows who Carina's mother is. He remembers her, remembers that Barbossa knew her. Carina's mom wasn't a whore Barbossa had a one-night stand with... she was someone he loved, that he cared for, and didn't hide it. Notice too that Jack, for all his talk, doesn't insult her. He knows she was someone special to Hector.
When Barbossa realizes Carina is his daughter he looks at the compass... and it points to her. The compass points to what you want the most, even if you didn't realize it. What he wants most is to be a part of his child's life.
The reunion of Will and Elizabeth. They see each other from a distance and start to run towards the other. Elizabeth with a look of disbelief and Will smiling with joy. They briefly stop and look at each other before embracing, at which point Elizabeth smiles and Will looks relieved. Then, after twelve years of separation and celibacy, they kiss passionately. One can only imagine how many times either of them dreamt of that moment and woke up just before they could touch.
While doubling as a funny moment, it's pretty heartwarming to see the pirates being non-judgemental during the scene where they misinterpret Carina stating that she's a horologist for her stating that she's a whore.
Carina: I am also a horologist. Jack: Nothing to be ashamed of, love. We've all got to make a living somehow.
The Black Pearl was cursed and shrunk to the point one could hold it in their hand, and sealed in a bottle where it rests on a miniature moving ocean. Even thought it is absolutely useless to him, Jack still keeps it with him over five long years, because he loves it so much.
The Pearl herself, when she is returned to her former glory by the power of Blackbeard's sword, brings herself to the surface in the start of the third act. She has been subject to shame and ridicule, being shrunk in a bottle for five years. As she comes to the surface, she shakes out her sails and struts, as if to say, "It's about time!"
For all they've been through and as many times as they've been at each other's throats, Jack finds a brief moment to mourn Barbossa's sacrifice in his own way.
"Pirate's life, Hector..."
Barbossa's death. He's gone from the cruel, monstrous undead captain of the Black Pearl who would gladly kill thousands just for the chance to eat an apple again, to a man who willingly gives his life without hesitation to save his only child. Knowing there's no chance of him coming back this time, he fearlessly leaps to deal with Salazar and takes the monstrous captain with him, simply watching Carina the entire time as he falls into the sea.
