If the government shutdown continues through this coming week, then the unemployment rate in the January report will be negatively impacted. This is a key week since it is the reference week for the BLS report (contains the 12th of the month). If the shutdown continues through next weekend, Federal employees who are on furlough will be counted as unemployed in the January report (CPS, Household survey).
However, the furloughed employees still have jobs, so their positions will still be counted in the CES (Establishment survey). So the headline employment number will not be directly impacted.
The closest example to the current situation is the October 2013 government shutdown that lasted from October 1st through October 17th (there are differences in what was shutdown). From the October 2013 employment report:
Among the unemployed, however, the number who reported being on temporary layoff increased by 448,000. This figure includes furloughed federal employees who were classified as unemployed on temporary layoff under the definitions used in the household survey.If the government shutdown continues, then the unemployment rate will probably bump up to 4.0% or 4.1% in the January report. Assuming the shutdown ends soon thereafter, the change in the unemployment rate will be reversed in the February report.