DropBox does not automatically rotate photos. Flickr and other photo share sites and phones rotate the pictures based upon how the picture is oriented i.e. the sky will always be shown on top. Dropbox will display pictures based on how the camera is oriented, i.e. the top of the picture will always be the top of the camera. If you turn the camera 90 degrees sideways for a picture, Dropbox will display the picture turned 90 degrees sideways.
Dropbox reads the EXIF metadata of the file and is not able to modify it. Therefore if the file is in a particular viewing position, Dropbox will also show the file in the same position. Reference: https://www.dropboxforum.com/t5/Dropbox/Rotate-Images-inside-the-Dropbox-APP/idi-p/149793
As a workaround, there are several web-based utilities that are able to modify the photo rotation stored in the photo's EXIF data, one of which is: https://www.thexifer.net/#dropbox. It allows you to link your Dropbox account and update the orientation within the photo's EXIF data.