More detail from ASCD:
1. What’s the bottom line for schools and districts? How would a government shutdown affect daily operations?Most schools and districts are unlikely to feel immediate effects of a shutdown because the advanced funding nature of federal education spending means that states and districts have already received much of their federal funding for the school year. In addition, the vast majority of school funding (about 90 percent) comes from state and local sources. Moreover, the U.S. Department of Education has awarded dozens of competitive grants in the past several days so that it is not held up by a shutdown.
2. Will any education programs be affected in the short-term?Head Start (which provides early childhood education to low-income families) and Impact Aid (which helps fund school districts that cannot fully rely on local tax revenue, such as those on military bases or tribal lands) depend heavily on federal dollars that are not necessarily distributed at the beginning of the school year. Thus, these programs could experience more acute and immediate shutdown consequences. This is especially concerning because Head Start and Impact Aid have already deeply felt the effects of sequestration. More than 50,000 children have lost access to Head Start and many Impact Aid districts have been forced to eliminate positions and programming because of sequestration.
More detail at the above link and at EdWeek's K-12 blog