Why? Copy FROM clipboard would be, sure. Copy TO clipboard... OK, I can come up with scenarios where it'd be a problem, but they're pretty far-fetched.
Run a timer overwriting your clipboard every 10ms. Prevent you from copying anything off a webpage and instead replacing it with a copyright notice. Etc., etc., etc.
Flash has offered these same clipboard APIs for years and these clipboard "attacks" have never been a problem before. I don't see how replacing Flash clipboard APIs with HTML clipboard APIs will change web developers' behavior.
(Github project pages have a flash application to handle this)