I verified that the API successfully deleted the files from Dropbox by logging in to the Dropbox site. However, they remain on my local machine long after deletion.
Clicking on the icon in the menubar indicates that it is up to date, but the icon in Finder is overlayed with Dropbox's syncing icon.
In case it's relevant, I'm using the Core API's PHP SDK (v1.1.3) installed via Composer.