per law, furloughed employees are not allowed to work, even voluntarily. It is against the law. So no, civilian teachers will not be teaching classes.
To clarify a few other things, military personnel are all required to go to work, but they will not recieve pay until an agreement is reached. At such a time, military will recieve backpay for those times. furloughed employees on the other hand will not recieve any pay during that time.
As for scraper's comment, i would bet many cadets don't do hw tonight with using the shutdown as an excuse, just because cadets use any excuse they can to put things off (plus, its Friday, and we gunna have a ball tonight).
Now each part of the government approaches it different. Here at AFIT (Air Force Institute of Technology) all of our classes will be cancelled until civilians return.