On Android, KeePassDroid registers as a keyboard to prevent other apps from retrieving your passwords from the clipboard. Not super familiar with iOS, but it seems like a good practice anyway, independent of its utility in circumventing asinine "no pasting" policies.
1Password does too but I've found that the Chase app and the Google Account manager all fail silently when using the keyboard to paste in your password. I called up Chase and they had no idea about why it wasn't working, although they could see my login attempts.
I only was able to figure it out when I changed my gmail password to something stronger and couldn't log back in and had to google the problem.
