If an author writes a book about a public figure, like, for example Tonya Harding, can the book author then put on Tonya Harding's picture on the cover of their book without Harding's permission?
If the image is a selfie, you need the permission of the subject. Otherwise, you need the permission of the photographer (or the person, if he hired a photographer to take his shot). You might edit the actual question to narrow the scope of the question.
Apart from copyright, there is a concern over "right of publicity", which could limit use of the image. This right prevents the unauthorized commercial use of an individual's name, likeness, or other recognizable aspects of one's persona, First Amendment notwithstanding. See this summary. A public figure is not exempt (unless that state's law is so written). Take California as a strong case of a right of publicity: you cannot commercially exploit a person's image, even if they are a famous actor (plenty of them in California):
Any person who knowingly uses another’s name, voice, signature, photograph, or likeness, in any manner, on or in products, merchandise, or goods, or for purposes of advertising or selling, or soliciting purchases of, products, merchandise, goods or services, without such person’s prior consent, or, in the case of a minor, the prior consent of his parent or legal guardian, shall be liable for any damages sustained by the person or persons injured as a result thereof.
This is all state law v. 1A, and in many places hasn't been litigated to a point to generate useful precedent but....
There's mostly a recognized difference between editorial usage and commercial usage. Even advertising an editorial product can be protected editorial usage.
Read Namath v. Sports Illustrated as well as Montana v. San Jose Mercury News.