Upgrade BI server to 2014 and stay on 2008r2 with the database engine server
Edit

If your setup involves your BI server being standalone and the report server database is being hosted on the BI server and not the production database then yes upgrade to 2014. You should consider looking at the new deployment methods with SSIS, very cool stuff.

If your setup involves:


your BI server only houses the Integration Services and Report Server binaries
your SSRS report database is being hosted on the production server
your IS packages are file system


Then the below would apply.



With SSRS that would be a supported setup if you installed SQL Server 2014 SSRS on your BI server. You can view the documentation here that shows the supported database instances to house the SSRS databases. 


  SQL Server is used to host the report server databases. The SQL Server Database Engine instance can be a local or remote instance. The following are the supported versions of SQL Server Database Engine that can be used to host the report server databases:



SQL Server 2012 
SQL Server 2008 R2 
SQL Server 2008 
SQL Server 2005


SSIS 2014 is another beast though if you want to actually use the new features that come with it for management: SSISDB Catalog or Project Deployment (which requires the SSIDB Catalog). You would have to have a SQL Server 2014 database instance in order to configure all this. The only thing you would be able to do with this type of setup would be a file system deployment, anything else I think is going to require the database engine to be at the same major version.