I'm following the instructions to set up App Insights to spool to SQL using Azure Stream Analytics, but I'm trying to deviate slightly to use an on-premise SQL server (that the web application already uses) over VPN.
At the point of adding the output, this is failing with:
Is it the case that IP addresses are not supported, or is it something more fundamental than that?
You are probably looking for answers directly to your question, which Jean-Sébastien answers succinctly. But an alternative architecture, if you haven't considered it already...
You could stream to a transient Azure SQL Database or Blob storage (likely cheaper depending on your workload), and then use Azure Data Factory tunnelled via a Self-Hosted Data Factory Integration Runtime to "send" the data back to on-premise SQL.
Data Factory V2 also has blob triggers, so rather than needing a schedule it could pickup any new blobs in micro batches.
I say "send" in quotation marks as the Integration Runtime actually creates an outgoing connection to from on-premise to Azure, yet gives the capability for push-like data transfer.
If data factory proves useful, here is a guide creating copy pipelines: https://learn.microsoft.com/en-us/azure/data-factory/tutorial-hybrid-copy-portal
Albeit this guide is for on-prem sql to blob, but it gives you a stronger starting point.