Albert's speech about being an adamant pacifist, particularly at the end when he says "I love you, Sheriff Truman."
The scenes where Ben Horne looks back on his childhood are both heartwarming and funny.
The night Catherine comes back home to Pete in her Tajemura disguise. The look on his face when he realizes it's her made this troper crack a smile from ear to ear.
In Fire Walk With Me the final scene where a despondent Laura, trapped in the Black Lodge, is visited by the angel that she thought had abandoned her. Her sudden realisation that she was still a good person worthy of love and protection, despite the terrible experiences she had suffered, was a truly heartwarming moment.
Major Briggs telling Bobby about his vision of Bobby's future happiness.
Cooper guiding Leland into heaven, and then telling Sarah how peaceful his death was.
Cooper's would-be exit from the town, with Harry giving him a fly lure and a Bookhouse Boys badge, and his perfect sendoffs to Hawk, Andy, and Lucy.
Despite Audrey being an eighteen-year-old girl with a big crush on Cooper (and admittedly quite sexy), Coop never treats her with anything but the utmost respect and kindness. When she attempts to seduce him in his hotel room, he gently turns her down, and offers to just be her friend instead, sensing that a friend is what she really needs right now, and leaves to get some ice cream for them to share before they have a talk about Audrey's troubles. The scene is easily the most heartwarming in the series.
Many of the scenes between James and Donna can be this, especially when the latter comes to realize how much she loves him.
Cooper in general is a heartwarmingly good and loveable character. The same can be said for Garland Briggs.
Real-life example: despite the horrible reception of Fire Walk With Me by critics, many actual survivors of incest came forward over the years to thank Sheryl Lee for her portrayal of Laura and what she goes through in the movie.
Meta one: David Lynch has agreed to continue making Twin Peaks! So maybe the Downer Ending will be averted...