Mountain lions do not carry prey up trees, in answer to the question. After killing prey mountain lions may begin to eat immediately or they may prepare the carcass. For example pumas usually pluck birds before eating them.
Pumas normally consume all edible parts of the carcass leaving the stomach contents, hooves and bones. When prey is abundant they may feed lightly leaving some of their prey.
Sometimes pumas intend to return to a carcass, in which case they may cover the remains with leaves, snow, sand, grass or whatever other substance or items are available.
Mountain lion kills are often scavenged by other animals including bobcats, foxes, coyotes, bears, pigs and wolves. Scientists are not sure what percentage of carcasses are scavenged but losses can be high. For example in Idaho 79/100 puma kills were scavenged by coyotes. Sometimes coyotes cache the remains of deer or elk carcasses within 24 hours.
The question in the title probably derives from the activities of the leopard. For example, in parts of Africa, because there are competitors such as lions, hyenas and wild dogs, leopards are forced to take their kills into trees. Any leopard kill made in open habitat is going to be detected by vultures which encourages the presence of other scavengers. In parts of the Masai Mara and Serengeti were competition from hyenas and lions is common, leopards frequently take their kills into trees to protect them.
The cougar does not do this notwithstanding that these two large wild cat species are similar in size. It may come down to competition. Arguably there’s more competition across the distribution of the leopard compared to the Puma. The mountain lion does not have to compete with larger predators such as lions and tigers or groups of lesser predators which are very formidable. Also the leopard is arguably a slightly more confident climber.