The code has been accessible for a while now, albeit still licensed as proprietary. I went through it a few weeks ago. The Winforms code depends heavily on GDI APIs, which AFAIK are windows only and closed source
BTW if Microsoft chooses to go opensource one day they can just start improving Wine together with CodeWeavers, ReactOS team and the community to the point when it could replace Windows completely. This is approximately the way they have chosen to go with their web browser switching to the Chromium Blink engine.
I bet Microsoft has an absolutely killer win32 regression suite. If we're devolving into pipe dreams, I'd probably want that even more than the actual win32 code.
There's an open-source implementation GDI through WINE, so it should be possible to get Microsoft's implementation of Winforms running natively on Linux.
It's not just GDI. WinForms is literally just a wrapper over stock Win32 widgets, for the most part. So you'll need the entirety of Wine, pretty much, and your app will look like a Windows app. At that point you might as well just run it via Wine.