Rental shop chain TSUTAYA reported the 10 most rented Japanese movies in their chain from 2016 January to 2016 June. According to it, the most rented movie was "Bakemono no ko (The Boy and the Beast)"
The following list shows the top 10.
The Boy and the Beast
Our Little Sister
Attack on Titan
UNFAIR the end
The Big Bee
Source is TSUTAYA club magazine, 2016 August.