When Max and Goofy have already gone over the waterfall, and Goofy's fishing rod's handle has just broken, sending him down into the abyss, Max whirls the broken rod around, copies Goofy's "Perfect Cast" absolutely flawlessly and ends up saving his father. The resulting pride brings Goofy to tears and leaves him all but speechless.
The montage where Max lets Goofy choose the stops on the road trip, because his dad was taking him where he wanted to go up to that point. Specifically the part where, after seeing Goofy having a hard time at the monster truck rally, Max decides to take him to a house of yarn.
I love Max and Goofy moments, but what about all those cute scenes with Roxanne? Funny/heartwarming all the way.
Early on, Goofy mentions mambo king Xavier Cugat and even does a dance, embarrassing Max. Later when Roxanne invites Max to watch the concert at her house, how does he react? Yells "Everybody mambo!" and does the same dance. Guess the apple doesn't fall too far!
When Goofy and Max appear onstage, Powerline, whom was obviously surprised that they were there, doesn't complain or call security like most performers would do, instead he lets them dance with him and seemed pretty impressed with Goofy's moves.
When Max gives Goofy the X-Games trophy, telling him to read the inscription, which is a Call Back to the first film: "I might not be your little boy anymore, but I'll always be your son." (Cue big hug between the two)
In An Extremely Goofy Movie, Goofy hands down Max an heirloom, an old-fashioned counting machine. Even though Max is less than impressed with the gift, he still thanks Goofy and says it means a lot to him.
For Goof Troop fans, the love stories in the sequel may both qualify. Goofy, who had been a single father for years, found a girlfriend right at the time that his son left the nest, so that Goofy won't have to be lonely in the following years. And PJ's Earn Your Happy Ending was a significant payoff to anyone who wished he'd just catch a break. Actually, his Character Development itself might count as a Heartwarming Moment, especially if you were familiar with the chestnut speech in "O R-V I N-V U", which is listed on the Heartwarming Momentspage for Goof Troop.