Dante was inspired by the Xoloitzcuintle breed (or "Xolo") the filmmakers played with during the film's research process, which is the national dog of Mexico. According to Lee Unkrich, the Xolo is featured in legends of the Aztec afterlife where the traveler needed to have a Xolo with them to make the journey through the afterlife.
Dante's tongue hangs out on the side of his mouth. In real life, this is because the Xolo is genetically prone to losing teeth and their tongue will hang out when this happens. When learning this, the animators decided to make this trait part of Dante's character.