GitHub has finally introduced a mobile layout for repositories, issues, and some other views (most of what you'd want to access on a phone). I especially like the "less fancy, more fast" approach – they've not just tweaked the layout to make it more readable on a small screen, but also dramatically reduced the page size, so it can be loaded more quickly on a slow connection.
Obviously, there are a lot of things on GitHub that just don't make sense to do on a phone, but I actually find myself using GitHub on my iPhone pretty often, e.g. reviewing the code of a new pull request, or doing some initial research on a third-party library I might want to use.
There has been a native app for managing issues for a while, but I didn't find that very useful – in most cases I just used the Mail app for reading them, and I rarely want to respond to an issue from the phone. The new mobile pages also have the big advantage that they work directly in a web view of any other app, say a Twitter client.