Will there be a new Lord of the Rings book?

In the original series, no, there won't be another book, because the author (Tolkien) died, but there will probably be plenty more Lord of the Rings-related books by different authors.

There cannot be any as the Tolkien estate tightly controls the copyright. However his son Christopher has edited and published many of his father's unfinished manuscripts of Elvish Mythology & History. The first being the Silmarillion.