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SQL Server drivers are backwards compatible, so if you can connect to 2016, there should be no issues with connecting to 2008 R2. That being said, my guess is that your problem is a configuration issue on the SQL Server side. You'll need to have a DBA check to make sure the named pipes protocol is enabled. You may also need to check to see if there are any firewalls that may be blocking network communications.
Here's a guide to troubleshooting these errors: Troubleshoot Connecting to the SQL Server Database Engine | Microsoft Docs
In addition to Ken's comment, also check your tableau server has access rights to the second server - more detail here https://onlinehelp.tableau.com/current/server/en-us/runas.htm
Following up on this one. Let us know how we can help further.
You were correct that it had something to do with a firewall. Thanks!