A surprising number, but an especially good one is when Paul resurrects Graeme, despite the danger to himself.
Before that, Paul took a slight detour during his grand escape, while the government agents are closing in on them, to stop at the home of the little girl he first met when he crash-landed (on her dog, which he apologized for...). And he brought back her teddy bear. Aww...
Especially heartwarming is when she talks about being ridiculed by the other kids when she was a little girl, the camera zooms in Ruth, and you can tell that the former religious zealot almost surely knows how she feels, having shared a similar experience while growing up.
There's some Fridge Heartwarming here, too, mostly Fridge since it's not something that immediately comes to mind given Paul's personality. But that he brought back Tara's teddy bear implies that Tara gave it to him as comfort when she was taking care of him. She does mention caring for him for days 'before the men came', and Paul, at this point not exactly having 60 years of experience with humans and earth in general (even though he was on a science mission) may have been scared shitless. And ere's this little girl taking good care of him, and he returns the favor when he finds out his staying with her ruined her life. Aaaaawe.
A small moment, but heartwarming all the same, when they are trying to escape from Zoil after Tara's house blows up and Graeme is running to catch up to the RV; he shouts 'Don't leave me!" After pulling him in, Clive smiles and answers "Never." Awww.
When Paul heals Ruth's eye.
Paul bringing Tara with him into space and "a new life" because he ruined her old one.
The reason why Agent Zoil is chasing the group: the simple fact that Paul is his friend and he wants to help him return home.