Is it possible to completely remove an issue from the GitHub issue tracker?
Still impossible. Another workaround to the ones suggested in the other answers is to label the issue as "deleted" (or any other label you might fancy better), to be able to filter them out if you use the github API to retrieve them. Obviously you should use that specific label only for this purpose, setting the label when you close the issue.
No, the github API only allows you to open/close/reopen issues. Here's the Issues API docs.
For posterity: Deleting issues would be a bad thing, since in general they can be targets of associations on github.
But if you are willing to sacrifice the collaboration info, here is a "whack it with a sledgehammer" approach:
I imagine this could potentially lose a lot of other linking information as well such as forks, pull requests, etc.
Update Nov 2018: You now can delete your issues!
See "Github - remove issues entered in error"
At May 2018, original answer:
Three 8 years later, and closing issues remains the answer (still no deletion possible).
See "The Ghost of Issues Past", where GitHub advise to check and close: