I am an American, and I believe in the fundamental ideals that led to the formation of this country. Because of this, I believe that flag burning, as political speech, is not only appropriate but should be legally protected. The one issue I have with this is that burning a flag has the potential to start fires or injure people.
As for the burning of books, it would depend on whether the books were being burned to make a political statement, or as censorship. I don't believe in censorship (and I don't believe that anyone who is a member here believes that censorship is, in general, a good thing). However, book burning to make a political statement should be allowed, so long as the information found in the books is replicated elsewhere and is available to interested parties. Otherwise, the book burning would be de facto censorship. Of course, precautions should be taken so as to prevent the injury of others or loss of property.
I did not consider apolitical motives for book/flag burning; my opinion on these motives would depend on the circumstances of the burning.