Bob and Helen Parr in The Incredibles, though this is played with a little. A lot of the movie focuses on the hitches in their marriage that come as a result of their differing reactions to leaving the superhero life (Helen is able to handle it, while Bob basically has a midlife crisis). Later, Helen is absolutely devastated to think that Bob is cheating on her. When they reunite and finally are open with each other though, Bob refers to her as "the most perfect woman in the entire world", and they are shown much more happy and adjusted once the rift between them closes.
Shrek and Fiona have their problems to deal with, but they love each other. Same goes for Donkey and Dragon.
The Rio franchise has 3 happy (most of the time) couples: Linda and Tulio, Blu and Jewel, and Rafael and Eva.
Carl and Ellie in Up definitely qualify, even if they had trouble making a family and fulfilling their dream to go to South America since by the time they were able to afford the trip Ellie soon passed away. This is demonstrated at the end, when Carl looks through Ellie's Adventure Book one last time to find the message "Thanks for the adventure! Now go have a new one."
Also the case in Cinderella 3. Heck, the first song of the movie has Cinderella telling us how happy she is in her married life and the prince several times referring to her as his "perfect wife". The sequel for Sleeping Beauty shows us that Aurora and Philip are quite happy as well.
Tarzan got an animated series where we see Tarzan and Jane are happily married. Jane even proudly admitted it to her high society friends when they came for a visit.
Rapunzel's parents in Tangled. One of the first things in the movie is the king (and most of his subjects, for that matter) searching furiously for a way to heal the ill queen. Even after their baby daughter is kidnapped, the two are shown supporting each other in the hope of finding her again. Rapunzel herself and Eugene are officially this after Tangled Ever After.
King Fergus and Queen Elinor in Brave, crossed with Perfectly Arranged Marriage. While Elinor admits at one point that she herself was uncertain about her arranged marriage, she now is very happy with her husband. On his side, Fergus was shocked to think that Elinor was ever uncertain about marrying him. Later, when he suspects that Mordu broke into the castle and attacked Elinor, he goes from his normally pleasant demeanor to a berserker rage.
Roger and Anita in 101 Dalmatians. The loss of the puppies (and later Pongo and Perdita) never puts a strain on their relationship. Pongo and Perdita themselves also qualify; like their owners, the loss of their puppies never affected their relationship, and they even team up in order to find them.
About halfway through The Prince of Egypt, Moses marries Tzipporah, and the two stick together through thick and thin for the rest of the film, with Tzipporah deciding to come with Moses to Egypt even if it would mean leaving her family behind forever, as well as rushing to help him when he's knocked into the dirt by the angry Jews.
The spiritual sequel/prequel/whatever, Joseph King Of Dreams has Joseph marry Asenath. The two are shown working together on plans to save Egypt from famine, have several children, and see their life together as proof of a better future ahead. When Joseph imprisons one of his brothers later, Asenath's reminding him of their earlier relationship was able to sooth him for a short period of time.