I have always been puzzled by this phrase. Who uses a comb on their teeth?
I just found the answer and thought I would share it with you!
I found it in a marvellous new book called The Grammar Cookbook. It's a grammer reference book basically but it's fun and interesting. If you go to the web site (trackerpress.com) and click on Sample Pages one of the pages is the explanation of the "fine tooth comb" mystery!

Robbie