Vulnerability in Server from SQL Server 2008 R2
The web components are/were used for the cube browser in SSMS (That changed in SQL Server 2012 IIRC), so you could probably remove it if you are not using that functionality and only browse the cubes from other clients, but that's an untested configuration so who knows for sure. It will definitely be unsupported.

As you state:


  Can we remove that component without breaking SQL server engine


If all you care about is the SQL Server engine (and not the Analysis server) you could consider removing Analysis services altogether.

However, I don't see why you would want to do that as according to the documentation this is fixed in SQL Server 2008 R2 by simply installing updates for Office.

Also according to that page it doesn't seem like it's the office 2003 web components you have installed with your version of SQL Server but the Office 2007 web components as the link for SQL Server 2008 R2 is pointing to the update for Office 2007

If I were you I would prefer patching the system than taking the risk of removing a component that came with the installation.

And as vonPryz mentioned in a comment. It's about time to look at upgrading to a more recent version of SQL Server if you are worried about support anyway.