We receive the notice, lock the repo, and inform its owner why it's been locked. At that point, they can either remove the infringing code or make a counter claim and we'll unlock it provided the original party doesn't file a court order.
More info: http://help.github.com/dmca/
After that it would take a court order to get it taken down again, but that would mean the repository owner will be liable for the damages, not github.
This is the one part of the DMCA that actually seems to work reasonably well. If you're confident that your stuff does not infringe then it's back up within a very short time and you know you're on the hook for any further fall out.
(IANAL)
As someone who runs a site that hosts content, this is great for us. We aren't allowed any sort of leeway in interpreting the claims and in return we are exposed to no liability. We are simply the middle man and have to do what's asked of us.