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
Web.config files do not apply to Function Apps and will indeed (as you found) be ignored.
Because of this limitation, features that require web.config customization are not supported with the standard Function App model (as you can see in the answer linked by @Mikhail, there are workarounds), and the alternative, currently, would be to handle this requirement in code.
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.