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.
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!)
In 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.
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!