Can we use a book figure/illustration in a review article and cite the source without getting permissions? I included two figures from book in review article. I emailed the editor and author of the chapter for permissions, but they didn't reply. I also mailed CRC publisher for permissions, but still didn't get response from them. 

If I still use those figures in a review article and cite the original source, will it be legal or not?
You cannot publish somebody else's copyrighted material in your own documents without a license.  Typically, the copyright is held not by the authors but by the publisher, e.g., CRC Press.

For most large publishers (including CRC Press, which appears to be owned by Taylor &amp; Francis), they have a standard procedure for obtaining permissions for republication, e.g., to include a figure as part of a review article as you wish to do. Depending on the particulars, sometimes it will cost money and sometimes it will be free.  In any case, sending email will generally be pointless because they will expect you to go through the standard permissions process instead.

In this case, if you go to the CRC Press permissions page, it leads you to a general copyright licensing service, which has a "pay per use" system for obtaining, among other things, a license to:


  Republish an article, book excerpt or other content in your own books, journals, newsletters and other materials


Finally, note that it is sometimes worth checking whether an alternate free version exists.  For example, authors may have also previously included similar material in a US government report or other item where you can freely use the material (always with appropriate attribution, of course).