Can I use SSRS 2016 with SQL Server 2008 R2 database engine?
You can use SSRS 2016 and fetch data from another databases.
When you defines the connection string of data sources, you can use whatever of your installed drivers.
According to MS Docs:
Built-in data extensions

Default data extensions in Reporting Services include the following types of data connections:
Microsoft SQL Server
Microsoft SQL Server Analysis Services
Microsoft SharePoint List
Azure SQL Database
Microsoft SQL Server Parallel Data Warehouse
OLE DB
Oracle
SAP NetWeaver BI
Hyperion Essbase
Teradata
XML
ODBC
Microsoft BI Semantic Model for Power View: On a SharePoint site that has been configured for a Power Pivot gallery and Power View, this data source type is available. This data source type is used only for Power View presentations. For more information, see Building the Perfect BI Semantic Tabular Models for Power View.