We will migrate from Windows Server 2003 to Windows Server 2012. My question is if we can use Team Foundation Server 2010 on the new server, or if we need to update to TFS 2012?
It appears that Sharepoint Foundation 2010 doesn't support Windows Server 2012, and I believe SQL 2012 doesn't work with TFS 2010, but if you can live without Sharepoint and you can use SQL 2008 R2 (or prior), you should be fine.
Take a look at: http://msdn.microsoft.com/en-us/library/vstudio/dd578592.aspx
Notice where it says:
Because Team Foundation Server installs SharePoint Foundation 2010 with the standard configuration, you can’t use it on Windows Server 2012. If you want to install Team Foundation Server on Windows Server 2012, use the advanced configuration and skip the installation of SharePoint Foundation 2010.
Regarding SQL 2012, take a look at this: http://blogs.msdn.com/b/bharry/archive/2013/01/11/tfs-2010-will-not-support-sql-2012.aspx