IP Restrictions are possible for Azure Websites as is protecting the site with Basic Authentication..
Are these, or any other web.config techniques possible on Azure Fu
Its been some time and the answer to this has kind of changed. While its still not possible to directly do this with a web.config file, these platform features are now available in the Azure Portal.
If you navigate to: https://portal.azure.com
And then go to:
App Services -> (Your Func App Name) -> Platform Features -> Networking -> Configure IP Restrictions
You can configure IP restrictions here and it will apply to your function app.
Similarly, there are now several authentication options under:
App Services -> (Your Func App Name) -> Platform Features -> Authentication / Authorization
This doesn't include basic auth, but it does include a bunch of stuff.