I’m actually more interested in a factual answer, but since nothing like this has ever happened (to my knowledge) and is unlikely to ever happen, all we’re going to get is idle speculation, hence MPSIMS.
Imagine a small city (say, ~4,000 residents) that’s in a generally rural area. Call it Pleasantville, Missouri. The town is big enough to provide its own public school district (one school each for elementary, junior high, and high school); if it didn’t have a school, the kids would have to be bused half an hour or more away to nearby schools.
For whatever reason, over the summer of 2012 every last parent of every last child in Pleasantville decides not to send their kids to Pleasantville schools any more; they’ll be either home-schooled, un-schooled, sent to private schools, or whatever. Registration Day comes and not a single kid is enrolled in any school in the Pleasantville district. The first day of school comes, and only the faculty show up; there is not a kid to be seen.
What would happen? Would the faculty continue showing up for work each day, since that’s their jobs, whether their classrooms are full or empty? Would the school district be forced to close the school (and be out however many millions of taxpayer dollars they spent)?