The story HTML file generated using Twine is a web-application and Dropbox is not a web-application hosting site, even though many people keep trying to use it as such.
For a web-hosted version of your story HTML file to be able to access externally stored media files (such as your images) using URL's similar to those used by a locally run version the external media files must be stored in such a way that they are accessible using Relative URLs.
When you share one or more uploaded files/folders on Dropbox, it creates a URL for each item shared and each of those URLs contains a unique key within it's path. To access those shared files from within your story HTML file you wound need to modify each of the external media references within it to contain those unique keys.
I suggest you look for a web-hosting site to host your web-application if you want your Readers to be able to access the story remotely.