I am configuring SSRS (2008 R2) in SharePoint integrated mode to work with a SharePoint (2010) site that is on another server, do I also have to install SharePoint on the database server that is hosting SSRS? I should note that the SharePoint databases are also on the database server that is hosting SSRS.
You need to install SharePoint and SSRS on the same machine. This does not need to be the same machine which has the actual databases on it (in fact it usually will not be the same machine).