"Lest We Forget", about the reunion of two lost lovers is one whole episode of heartwarming.
"Don't Say Goodbye": A friend of the real Amy Crane took over Amy's identity after she had passed away, rather than tell her long-lost grandmother Agatha the truth, and had helped take care of Agatha ever since.
Amy: It's not Amy, Agatha. I'm Carol. Carol Foster.
Agatha: Oh, dear.
Amy: And I am not your granddaughter.
Agatha: Well, perhaps not of my blood, but you are my granddaughter.
Amy:(sobbing) Why didn't you say anything?
Agatha: Because all my life I dreamed about having a granddaughter.
Every time Higgins expresses (and attempts to hide) his concern and fondness for Magnum. "Aw, Higgins, you were worried about me." "Nonsense!"
To get an idea of how much they've warmed to each other, compare earlier episodes like "J. 'Digger' Doyle" (where Magnum and Higgins really don't get along) to later ones like "I Never Wanted to Go to France, Anyway...", where at the end, Higgins, without hesitation, tries going back in to rescue Magnum from the fire set by the villain. It's only when Magnum's shown to have already escaped that Higgins stops.
Higgins' interactions with Zeus and Apollo are likewise revealing. He cares for the lads as if they were the children he never had.
"Rembrandt's Girl": Despite being trapped in a bank vault for the better part of a weekend, Magnum and Susan finding they have a lot in common and eventually adorably falling asleep together definitely counts. The great chemistry (and, at the end, unique laughs) of Tom Selleck and Carol Burnett is at least partly why.
The final scene of "Going Home" crossed with Tearjerker, where Magnum visits the Vietnam Veterans Memorial to say goodbye to his half-brother Joey, who was killed in Vietnam, while Joe Cocker's rendition of Bob Dylan's "I Shall Be Released" plays, perfectly setting the mood, especially when Magnum says "Goodbye, Joey" in the end.
"Mad Dogs and Englishmen" is one of those times where Higgins doesn't even try to hide his fondness for Magnum. Saying he's leaving the estate for good, he gives Magnum one of his belongings (a popsicle stick bridge).
Higgins: Please take it as a gift. Magnum: But I don't have anything to give you. Higgins: You have been my friend.
Zeus and Apollo are devastated by Higgins leaving and just lie there on the beach, so Magnum tries to cheer them up by getting them to chase him. They don't move a muscle, but later, they begin following him around the estate and refuse to leave his side. Magnum is both irritated and sympathetic.
It's extremely crossed with Tearjerker, but Magnum's letter to his daughter Lily in "Unfinished Business", especially his last line, after letting Quang Ki go so the American P.O.W. in his prisoner exchange can come home.
Magnum: And so, Lily Catherine, even though I know you will never read this, I needed to write this to tell you I once did something I think would have made you proud.
Once he is finally reunited with Lily in "Resolutions", Magnum happily takes on the role of proud, doting dad pretty much immediately.