What happens to the person who owns the property when the property tax lien on their home is auctioned in a public auction?

It depends on the jurisdiction. You must check your state laws. There may be a right to redeem: you pay the government the taxes, penalties, interest, fees and any other costs, and the person who bought the lien is reimbursed. Redemption periods vary according to state law and those time periods are critical - if you miss it you could lose the property.

Many jurisdictions now use a judicial process to take and sell property for delinquent taxes. Redemption is barred by the decree. Massachusetts is one example. In that case the buyer is the new owner and you must move out.