I was furloughed because of the government shutdown. Do I still have to pay my chapter 13 plan payments?

Yes.

The "confirmation order" that the bankruptcy judge signed when your chapter 13 plan was approved is a court order that commands you to pay the payments.

Of course if you don't have income and cannot pay the payments, the court would not try to hold you in contempt of court. And hopefully the shutdown will not last long.

But if you cannot pay the payments, and that continues for a period of time, your trustee will file a "motion to dismiss" your case for non-payment. If that happens, you'll have to either bring your payments current, file a motion to modify your plan to excuse the payments, or allow the case to be dismissed and re-file another case later when you again become able to pay.

There may be other options in your case; consult your bankruptcy attorney.