The IRS has issued a notice on their website (“IRS Update on Shutdown Impact on Tax Court Cases; Important Information for Taxpayers, Tax Professionals with Pending Cases”) dealing with issues that arise with taxpayers for whom the shutdown has created issues with their tax situation.  Although the shutdown has now ended (at least temporarily), many issues will continue for quite a period after operations resume at the Tax Court.

The IRS notes that the government shutdown did not stay the accrual of interest on taxes that may be determined eventually to be due on matters before the Tax Court:

To avoid additional interest on the tax that you are disputing in your pending Tax Court case, you can stop the running of interest by making a payment to the IRS.  Go to www.irs.gov/payments for payment options available to you.  The IRS is continuing to process payments during the shutdown.

Another problem is that the IRS computers continued to operate as they normally do when the government is not shut down, meaning taxpayers whose date to file a Tax Court petition fell during the shutdown may discover their tax has now been assessed by the IRS even though petitions sent to the Tax Court were being returned to taxpayers. 

The notice on the website provides:

If you receive a collection notice for the tax that is in dispute in your Tax Court case, it may be because the IRS has not received your petition and has made a premature assessment.  When the government reopens, the IRS attorney assigned to your case will determine if a premature assessment was made and request that the IRS abate the premature assessment.