Connect a database hosted on Amazon RDS with a private subnet to PowerBI?

When you have a database in a private subnet, you need to establish a secure and controlled way to access it from external applications or services like PowerBI. This is typically done using a Site-to-Site VPN Connection. You can try to follow this guideline Connecting the Microsoft Power BI service to AWS data sources.

This link provides AWS's recommended configuration for the setup you're interested in: Recommended configuration.