Answering my own question:
I'm using a Mac. Any folder on a Mac that has a custom Icon contains a hidden file called "Icon\r" which holds the icon. I have the project in a shared Dropbox folder. Dropbox automatically modifies the folder icon to indicate it is shared. Hence, the Icon file. See here for more details.
If you try to delete this file, Dropbox regenerates it. You can exclude it from the project and from Git source control, but you can't get rid of it as long as the folder is shared.