Hi
It will not stop running.But the 2003 server not under Support of ms,means could not install security&critical etc updates,this about server 2003 site.
Your 2003 wsus server will run.I strongly recommned that you should upgrade your server 2003 environment to newest version servers.(also 2012r2),for security reasons.
Also check detailed informaiton Windows server 2003 Support is ending july14,2015 article;
http://www.microsoft.com/en-us/server-cloud/products/windows-server-2003/