In this case, no the church does not have a US Constitutional Right to burn the books. A bit of history from Schenck v. United States (1919) will help clarify:
The question in every case is whether the words used are used in such circumstances and are of such a nature as to create a clear and present danger that they will bring about the substantive evils that the United States Congress has a right to prevent. It is a question of proximity and degree. When a nation is at war, many things that might be said in time of peace are such a hindrance to its effort that their utterance will not be endured so long as men fight, and that no Court could regard them as protected by any constitutional right.
- US Supreme Court Justice Oliver Wendell Holmes , Jr.
Source: http://en.wikipedia.org/wiki/Clear_and_present_danger
It is pretty much common knowledge the US is engaged in a war in Afghanistan whose population is majority Muslim. General Petraeus has denounced this act, saying it would lead to retaliations against US troops in Afghanistan and the Middle East. (Source: http://online.wsj.com/article/SB10001424052748703713504575475500753093116.html?mod=WSJ_hpp_MIDDLTopStories)
If this were during peace time when the US was not engaged in Afghanistan or Iraq, then the church would have every right under the 1st Amendment of the US Constitution to burn enough Korans to fill a Mosque. Since we are at war, they do not have the 1st Amendment Right of Free Speech to do it.
This case of the proposed Koran burning is a classic example of yelling fire in a crowded theatre. You just don't do it at this moment in time.
BOLD = corrected part
