Does the Bible have anything to say about government shutdowns?

Not directly.

But there is a Biblical principle which bears directly on the impact of such a shutdown, regardless of whether or not it is owing to one party or the other – or to both equally.

2 key passages come to mind: ““You shall not oppress your neighbor or rob him. The wages of a hired worker shall not remain with you all night until the morning.” (Leviticus 19:13, ESV) ““You shall not oppress a hired worker who is poor and needy, whether he is one of your brothers or one of the sojourners who are in your land within your towns. You shall give him his wages on the same day, before the sun sets (for he is poor and counts on it), lest he cry against you to the LORD, and you be guilty of sin.” (Deuteronomy 24:14–15, ESV)

The Joe-Average government employee who is impacted by the withholding of the timely payment of their wages has good cause to cry out to the Lord over it. It does not matter that their wages will be restored in due time. Many, if not most live from paycheck to paycheck. And irrespective of whether or not that is a right position to be in – their being deprived of timely payment for work rendered is sinful on behalf of all of those who are allowing it to happen. Republicans, Democrats, the President and the Congress need to find a different way to settle their dispute. It is shameful, un-Biblical and sinful to let this situation arise and persist.