At least in the case of Java package paths map 1-1 with file paths, right? Is Go similar? In that case, moving a function to a different package/directory would change the url? Would the old url break, or continue to show the old/stale version?

Yep, but those languages also define equality in their type systems using the package paths. It's tricky, of course. :)

We want to be able to track function moves and renames across packages or even repositories. It'll take some more work to get there.

Yes, certainly a hard problem. I'm just trying to understand what happens today. Does the link break or (better) just stop getting updated?

It will break if there is no revision, or a non-absolute commit ID (such as a branch or tag). An absolute commit ID link will continue to work.