Q. Does APA allow the title Dr. before the surname in a sentence, in the body of the paper (in-text)?

On p. 23 of the Publication Manual of the American Psychological Association, 6th edition, in regards to bylines, it states "Omit all titles (e.g., Dr., Professor) and degrees (e.g., PhD, PsyD, EdD)."

It would seem to make sense that this would carry over to the text.

And, this is speculation, but if needed, you might make reference in an early sentence ... "R. Lee Sharp, Dr. of Medicine at the American Cutlery Institute, enters the field of autopsy guides with this page turner." Then, is subsequent references she would only be referred to as "Sharp" ... "whether you work on a butcher block or a morgue slab, Sharp's book is the one for you."

I suppose you could also consider if the author's credentials are important to the work. It would not be for fiction, but if the book is on physical therapy, the fact the the author is a physical therapist would lend authority to the work.

 And finally, on p. 11 of the Publication Manual of the American Psychological Association, 6th edition, in regards to book reviews and other types of articles, it states "consult with the editor of the journal to which you are considering submitting the manuscript for specific information regarding these kinds of articles."

If you did not want to go as far as consulting the editor, you could take a look at other book reviews in the intended publication and see how they treat the use of titles.