No, English speakers do not use "bury" in those cases. I would say "Please cover your ears with your hands," "Please put this toy in your hands," "Let's cover this toy with the book." The reason I use "cover" and "with" instead of "bury" and "in" is because "bury" is only used for when an object is covered in something like dirt or covered by the ground, for example, I would say "Please bury the bone in the ground" to a dog, because "bury" is usually only used when something is underground and permanently covered by dirt or sand.

No, we wouldn't say these things. The expression only really works with 'face.' By the way this expression has a pretty specific meaning, you would only use it if someone was extremely upset about something, it's not a neutral way to describe someone covering their face with their hands. Put your hands over your ears. Cover your ears with your hands. Cover the toy with your hands. The third sentence could actually work but it doesn't mean what you intend it to. If you bury the toy in the book, the toy goes inside the pages of the book, not underneath it. So it would have to be a very small and particular type of toy for the sentence to make any sense. Otherwise: Let's put the toy under the book.