What happens if the property Title Insurance Co goes out of business?

A Title Company (the actual insurancer as in First American, Chicago, etc.) is regulated by a state's Department of Insurance. If the company dissolved, then they would have to follow the state's guidelines for closing. This is why a title company has to prove so much in reserves, liquidity, etc. to a state to even do business in a state. Typically, title companies are sold or bought out before they would ever go out of business. A Title Agency (the local agent who has an agency agreement with a title Company) must keep separate accounts for escrows/trust accounts. Ideally, if an agency closed its doors, all accounts would be reconciled and it could shut down with all Policies issued. If an agency shuts down and does not have the above issued addressed at the time of closing, the title company will complete the same, issue the Policies, etc. of the Agent/Agency or they may assign the same to another one of their Agents to be completed. Since ultimately it is the title company that is responsible for the issuance of the insurance, it is their responsibility to make sure that all policies sold by the agent are properly issued.