A title company's obligation is to its insured, the owner, the lender, or both. If there was an error on the mortgage payoff or the amount collected on the HUD, the title company has a duty to work out a solution and get that mortgage satisfied.
That does not mean that the title company will not pursue every means at their disposal to recoup funds owed by the mortgagor. So, whoever owed the mortgage money that is the subject of the error will eventually have to make restitution and pay up. It's their obligation.
The title company may also pursue a settlement with the mortgagee. If the lender gave a payoff letter that contained the wrong figures and the title company relied on it to close, they may be able to force the lender/mortgagee to accept the lower amount.
The main issue is that the title company goes to bat to protect the integrity of the title and will remove the mortgage lien by whatever means they choose.