The Dropbox iPhone app only stores local copies of files if they are marked as Favorites. To remove the file from your Favorites list and your iPhone:
Tapping the star icon while previewing a file will also remove it as a favorite.
To remove multiple files, you can tap the check mark icon at the top right of the Favorites tab, select the file(s) you want to remove, and tap the Remove button at the bottom of the screen.
To delete a file from your iPhone and all of your computers and devices simultaneously:
Tapping the trash can icon while previewing a file will also delete it.
To delete multiple files, you can tap the ellipsis (...) icon at the top right of the screen and then tap the Select option. Check the files you want to delete and tap the Delete button at the bottom of the screen.
Once you confirm your decision to delete a file, it will be removed from your iPhone, the Dropbox website, and all the computers linked to your Dropbox account.