They bury Finn on the Surface. They have a hard time locating anyone related to Finn, so they're the first person to be buried in the Monster Cemetery in New New Home. (Our king is amazing at naming things.)
The dummy heavily suffers the blow of Finn's death, but reconciles himself to it surprisingly quickly. His act of guiding Finn throughout the Underground makes up for his previous acts of cruelty to human children in the eyes of the monsters who knew about all that, and he is accepted back into their community. He joins up with Mettaton, Napstablook, and Shyren in their quest for fame in the Human World, and ends up becoming a heavily liked and honored news reporter.
Frisk - for that is the other human's name, the one who killed almost everybody - has no family either, and is adopted by King Asgore and Queen Toriel. Frisk claims to have been possessed the entire time, and nobody doubts them. Both they and Sans suffer heavily from PTSD for several years after the events, but they both receive undying support from their friends and family members and eventually grow to live with it. Frisks helps heavily with integrating the monsters back into life on the Surface, and is named Chief Ambassador between Humans and Monsters.
Papyrus gives up his dreams of becoming a Royal Guard and instead devotes himself to becoming a therapist. His passion drives him to succeed, and he ends up caring for and helping hundreds of humans and monsters alike. He writes several books on the subject and becomes a world-famous author.
Flowey spends the rest of his days being very confused, but well looked-after by his parents, to whom he finally confides the whole I'm-actually-Asriel secret. Though he never learns to love, he's deprived of his RESET power, and therefore never tries to amuse himself by harming others. Instead, he makes the most of his life on the Surface, helping Frisk and Papyrus with their various duties and occasionally subbing for third grade teachers.
Finn is always remembered. The monsters set up a great memorial for them, and they are mentioned in all of the history books written after that time period. "Finn the Great", "Finn the Kind", and "Finn the Innocent" are all titles used commonly to refer to them. They are remembered as the hero and the savior of the monsters.
The Angel of Life.