Can I use SSRS 2016 with SQL Server 2008 R2 database engine? I don't know that much about the state of products offering reporting/analytics, but I first got into OLAP in the late 90s, with a data warehousing tool, but then didn't touch it again for a while.  My vocabulary on the topic may not be precise.

I have data in a SQL Server 2008 R2 database.  The application putting data into it is not certified for SQL 2016, but SSRS 2016 (SQL Server Reporting Services 2016) has caught my eye.  It seems to have the features I'd like to use with this application.  I am adding reporting capabilities to a 3rd party application.

Can I create a new machine with just SSRS 2016 (which is a feature of SQL Server as a whole) and point it to the SQL 2008 R2 server with the database such that I can then start doing reporting on the application database?

What are the ramifications if the SQL 2008 R2 server is also an Enterprise server (or cluster) with many other databases on it?

This is not related to Sharepoint.  The 3rd party application is not a Microsoft application.
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.