The scenes between Charles, The Holy Roman Emperor, and Catherine of Aragon. You can tell that he truly cares for his aunt, and her happiness. There is a genuineness to his concern for her that goes beyond the political, when he is told that Henry is tiring of her.
Most scenes between Mary and Elizabeth. Specific ones include Mary comforting baby Elizabeth in season 2 (extra points since Mary has every reason not to care if this particular baby is upset), the two sisters snuggled in bed and talking about their new brother in season 3, and Mary telling Elizabeth that they've both been restored to the succession in season 4.
Any scenes between Katherine of Aragon and Mary or Anne Boleyn and Elizabeth.
Any scenes where Henry shows affection for his children.
Many of the scenes between Mary and Chapuys, especially in the fourth season.
At Christmas in Season 3, when Henry is presiding over the festivities with his new Queen, Jane Seymour, and his newly-reinstated daughter Princess Mary. Mary and/or Jane have arranged to bring his most-recently-delegtimized child, Princess Elizabeth, to court, and Henry publicly calls her his daughter (important as he'd had her mother executed for adultery and speculated Elizabeth wasn't his) and lifts her up to sit beside them. He announces to the court, "Je suis en familie (I am among my family)."
The relationship between Henry and his Fool, shown during the immediate aftermath of Jane Seymour's death. At first it just starts off with the Fool being, well, a fool. But as the episode progresses you see the Fool helping Henry through his grief until he ends up somewhere around being pseudo-paternal.
Henry: *tearing up, his head in his Fool's lap* I miss her, Will. I miss her so much.
Fool: *tenderly* I know, but this too shall pass. Why go on dwelling in darkness? You know, the land of the wounded king is only a parched wasteland, thirsting for rain and your majesty's grace? *Henry snaps out of his BSOD*
Each and every interaction between Mary and Eustace Chapuys, the Spanish Ambassador. Though Henry, at times, treats her properly, it is Eustace that is her true father figure. Every private moment between them speaks of nothing but compassion and care, despite Chapuys' ostensibly political purpose.
The procession of the King as he journeys north, with all the people cheering the royal family.
The relationship between Catherine Parr and her second husband. He genuinely seems to be a kind man who cares for her, sadly dying in pain, and she seems to genuinely care for him, though she also loves Thomas Seymour. It makes her catching the King's eye, and the loss of trust it causes, all the sadder.
At the end of the show, Henry sees Anne Boleyn (either as a ghost or his own imagination) and she tells him that although she desperately wanted to give him a son, she is now so proud of Elizabeth. More so, she's the only ghost that Henry begs not to leave.
The last meeting between Henry and Charles Brandon.
The fact that Anne of Cleves and Henry were able to remain friends despite having been divorced from each other.