Herbie himself and anything he does is bound to be heartwarming. He truly does live up to his name of being The Love Bug. Much like a dog, he wants nothing other than to be loved, and once he's met someone who takes him home with them, he remains extremely loyal to them, and even sometimes helps them find romance.
The Love Bug (Original Version)
In the first movie when Jim wonders why Herbie chose him:
Jim: What I don't understand is, out of the millions of people in the world, why it picked a dogeared second-rater like me.
Carol: You stood up for it once. I guess it thought you were...worth belonging to. I understand that.
(Note: They get married at the end, Herbie is The Love Bug after all!)
Herbie Goes to Monte Carlo
Herbie is trying to impress Giselle, but he's dirty and acting like such a lovesick goofball that she'll have nothing to do with him. Only when he cleans up, gets some flowers, and tries a more subtle approach does he succeed.
Herbie falling in love in Herbie Goes to Monte Carlo, especially near the end of the big Trans France Race where the plucky Bug rescues Giselle from being wrecked in a river. They then spend the night after the race watching the fireworks over Monte Carlo.
Diane and her Lancia Giselle, after staying in first, have crashed into a lake; the moment he sees them, Jim is worried and wants to go back in spite of the fact that he is in the middle of a comeback. Once Diane is rescued, Herbie goes in to pull Giselle out. It doesn't matter that either of them are in the middle of a race, they have their priorities straight.
It's a small moment, but as Bruno passes by everyone in the aforementioned moment he gives them a salute.
After Diane witnesses not only Herbie pulling her car out but refusing to leave, she doesn't ask questions or try to deny what she's seeing. Instead, she gives Herbie a speech to have him rejoin the race, asking him to win it not only for Jim and Wheelie but "for her", her Lancia. There's also the fact that she chooses to stay with her car (and names her Giselle) rather than take Jim up on his offer and join them in the race in Herbie.
Multiple times throughout the race itself, Jim, Wheelie and Herbie are knocked to last place due either being knocked off the track, redirected due to the diamond thieves, or having to stop to get the aforementioned diamond out of Herbie's gas tank. Jim is smiling every time they catch up. Considering that Jim hasn't had a win in twelve years, it's great to see.
The Love Bug (1997)
Given that the 1997 Love Bug remake is definitely one the most heart-wrenching films in the series, there's bound to be plenty of tender moments.
Herbie and Roddy's entire relationship. Roddy loved Herbie from the moment he first saw him. Throughout the first half of the movie, Roddy is the only one who Herbie can truly talk to, as he is the only one who knows he is alive.
When Herbie goes out looking for Hank, even though Hank was upset and wanted nothing to do with him. The way he honks his horn, trying to call out his name is just adorable.
After Herbie is rebuilt everyone is so worried that Herbie won't "wake up" when Jim starts the ignition that they take a moment of silence. Then the car starts and Jim asks if Herbie is there. Herbie honks in joy and everybody is so relieved!