Back in 1918, The Migratory Bird Treaty Act was passed to make it illegal (with a few exceptions) to possess feathers and nests because of the declining population of bird species.
Do you know one of the reasons so many birds were killed for their feathers? ..... to decorate women's hats! Another reason was because collecting eggs and nests was such a popular hobby.
Even though it doesn't seem fair not to be able to pick up a feather or fallen nest, it's too difficult for enforcement officers to prove whether a person had picked it up off the ground, or had killed a bird or pulled a viable nest out of a tree.
Places like nature centers and zoos can have feathers for education, but they must get permits from the U.S. Fish & Wildlife Service, and keep records of what they collect.
So even if you find a pretty cardinal or blue jay feather, don't keep it. You don't want to risk getting in trouble for owning an illegal feather (I put my feather back in the field).
Also, don't pick up and keep a dead bird. You can't even salvage a bird that died from disease, collision with a car, or from electrocution. Salvage permits are required because poaching puts many migratory bird populations at risk. If you find a dead bird or feathers that you think a repository might want, a bird that has been shot or you suspect poisoning, contact the Fish and Wildlife Service or a Game Warden.
The Fish and Wildlife Service has created an online Feather Atlas - a complete list of protected birds.