Best case scenario: owner donates them to a permanent collection, such as a university or botanical garden. The collection has to be of pretty high caliber for this, both in species diversity, rarity, and overall quality of growth/health. Sometimes, when an older person becomes unable to physically grow plants, they'll find interested friends to give plants and cuttings.
Often, however, a plant nut may pass unexpectedly. Not always but most of the time, this will immediately cause the plants in a collection to decline due to lack of care. In such a situation, houseplants are rarely the top concern of that person's next of kin. Most often, the children do not have the knowledge, experience, time, or desire to deal with the upkeep of a large plant collection. Eventually, the heirs of the collection choose to keep, sell, or donate it in pieces or in full.
My wife and I visit a considerable number of garage and estate sales almost every weekend, and resultingly I've seen almost every situation you can think of. A lot of plants wind up in the UGA collection this way.