Scheduled Refresh should work on Web APIs (Web.Contents in M)

If you apply a Web API using the Query Editor, and the API requires the use of POST with specific headers to return a successful response, when you try to deploy it on the service and turn on Scheduled Refresh, it will fail.

One of the common APIs that Scheduled Refresh fails to support is Microsoft Text Analytics / Sentiment Analysis, which can be developed in Power BI Desktop, but cannot be scheduled refresh by the service: https://community.powerbi.com/t5/Community-Blog/Sentiment-Analysis-in-Power-BI/ba-p/55898