You'll have to check with the vendor of the app, but from my experience the main data usage is when you actually download the book. If it's Kindle then I think it does some cloud syncing while you are reading to remember book marks, what page you are on, highlights, etc so there is probably small amounts of data being used to sync your progress to the cloud. That way when you open it on another device it's the same as you left off on the phone. This should be small amounts of data.
Not sure if you can turn those features off or not. Probably a better question to direct at the maker of the app if you need more detailed info.
Thanks!