Sure you can do some work locally but if your CI server can't check out the branch to build it and you can't approve pull requests for merge, then its a bit limiting.
Even if you run the CI on your own network, if it can't check out from Github it isn't going to have the latest code unless you're hosting your own repos for CI. Developers will need to push their changes to CI Git in addition to Github. Its certainly possible but far outside the practices of what most companies relying on Github are likely to do for the once or twice a year it goes down briefly.