It is really simple actually, and if you are having that IDM issue also then this is most certainly will be a relief for you. So below are the easy steps to perform this....
1. After login into your Facebook account click on that "chat" button on the lower right end of your screen (look at the screenshot below).
2. Then choose "options" which is located on the top of the window caption.
3. After clicking on that you'll see a another menu window from that click and remove that tick in front of "play sound for new messages".
Yep now you've successfully turned off the facebook new message play sound!.