I've installed Django support in Visual Studio Code and associated */templates/*.html with django-html as the extension demands.
However, it can't auto-compete HTML Tags as I've done so.
And if I just associate HTML with itself, then it can't intellisense Django Template code.
How can I autocompleting both?
I managed to do it by:
installing the django extension
adding the following configuration to my workspace config.json file:
"emmet.includeLanguages": { "django-html": "html", }This is essentially the same solution, but using the VS Code UI, which may make it easier for some people, especially if you're new to this and your settings.json file hasn't been generated yet.
In VS Code go to File -> Preferences -> Settings
Once there, you can switch to the 'Workspace' tab if you want this setting to be for this project/workspace only, or stay with the 'User' tab if you want this across all projects/workspaces.
Open the 'Extensions' item in the list and click on 'Emmet'.
Under 'Include Languages', click the 'Add Item' button. Complete it with:
Element: django-html
value: html
and click the 'OK' button.
This will add the settings for you to your settings.json file, or generate a new settings.json file if you don't have one. 
Note: For autocomplete/generate to work, you may need to type your tag without the brackets, e.g. li (not <li> ), then press Enter to get <li></li>
As namespace_Pt said, I tried it and it works. I will list which extensions are in my Visual Studio Code installation.
Django 1.2.0
Visual Studio IntelliCode (I tried, and it works without this one)
"emmet.includeLanguages": {
"django-html": "html",
}
I added it, at the end of the settings.json file. I find the file from the settings's search bar. Just undo what Visual Studio Code added itself and add the code above. Don't forget to add a comma.