The tax refund amount (or amount owed) that displays on your screen in TaxAct will not change once you receive the refund or pay the tax bill. Think of that as if you had printed a hard copy of the return - the static amounts that were listed on the paper don't change after you file, pay, or receive your refund.
In the same way, the amount indicated in your TaxAct program reflects the amount that was filed to the IRS. That information is for record keeping and not a "live" indication of what you owe the IRS or what they owe you.
If you owe tax, you will need to keep track (outside of the TaxAct program) of how much you pay until the full liability has been paid.