The comment feature is very handy on dropbox documents, but there is no functionality to export them, and no API to use them. Fortunately, it seems it exists an undocumented /comments2/list_comments API. I made a proof of concept of this API to get the ability to exports comments.
The HTML dropbox application uses a /comments2/list_comments API that gives us what we want. You need a proper authentication and the id of the file you want to see.
The application is very straight forward, uses the Dropbox Chooser to pick a file and allow the user to download comments. Below it a screenshot and here is the github repository with full source code and some documentation : https://github.com/rpeyron/dropbox-comments-downloader