As most will know, the US Postal Service is unaffected by the US Government Shutdown because they "receive no tax dollars" for their operation. So post offices are open and mail delivery is as usual.
However, it is interesting to note that the US Postal Service OIG (Office of the Inspector General) is affected by the shutdown as illustrated by this statement on their official website:

Also affected by the US Government shutdown is the Postal Regulatory Commission, the entity charged with reviewing the US Postal Service request for a 2014 postal rate increase.