TDF Takes costumes.
Build it Green takes recycled materials. Many shows donate stuff there.
Materials for The Arts takes recycled materials and puts them back into not-for-profit organizations. I remember seeing stuff from THE LITTLE MERMAID years ago after that show closed.
Sometimes the production management will send it BACK to the carpentry shop that built it initially, and they keep what they want, and reuse pieces on future shows.
sometimes backdrops will go into service at rental companies