I don't believe there are any 'push' notifications, but you can use the "threads...

shrikant on March 4, 2012 | parent | context | favorite | on: How I built a Hacker News mobile web app

I don't believe there are any 'push' notifications, but you can use the "threads" link on the top bar to see each of your comment threads and scan for replies.

Not ideal, but it's still better than either remembering to check the discussion itself.