The most comprehensive study is Modern Greek evidence for the ancient Greek vocabulary by George Pelham Shipp.
The standard work on the use of ancient Greek in Modern Greek is Andriotis, Lexikon der Archaismen in neugriechischen Dialekten.
Also useful is the Λεξικό αρχαϊστικών φράσεων by Δημήτριος Τσιρόγλου.