We have a Win2003 file server that is the heart of the org. Lots of UNC paths point to it. Drive letter mappings need to stay stable.
How can we abstract the UNC path, in such a way that we can move to a Win2012 server and preserve the UNC paths?
(Ideally, printer UNC references would also be preserved, but we could live with this changing/breaking)
It will take some transition work, but the first thing you should do is strongly consider using a DFS Namespace for your domain. DFS lets you have a single directory that links off to various file servers and shares underneath. You can easily retarget the links to a new server at any point, and this will be transparent to the user. But if you aren't already using DFS, then this will require making disruptive changes, but once you have made them you should be good for a long time.
The other thing that can help in a migration is to use a DNS alias to redirect the DNS name that used to be belong to the old server to the new server. This does take some work though. See this question.
Assuming you are not looking to do in place upgrades, most of the common roles can be migrated using the Windows Server Migration Tools.
There are other ways, of course but the migration tools are a supported method