I'm presuming you mean an open end line of credit, right?
When a title insurer is paying off an open end mortgage which is a line of credit, they should request that the mortgage lender freeze the line. With evidence of a frozen line, the payoff letter is reliable. If the mortgage lender refuses to freeze the line, even with a written request of the borrower, the payoff is kinda risky. We handle it by re-checking the payoff verbally before remitting.
In either case, we always write or type directly on the payoff letter "Please close and satisfy." and have the mortgagor/borrower sign it. Some mortgage lenders will provide a statement of that sort on their payoff letter and place for the mortgagor/borrower signature.
The title insurer has a duty to be careful in their examination of title and look at the mortgage document to see if it is open end and then follow the steps I have just outlined. Why? Well, if the line isn't closed and the underlying mortgage satisfied, the interest of the insured buyer or lender is at risk.
Notice I said buyer or lender and I also said insured. A title insurer is performing this function to protect those they insure.
If you are the mortgagor/borrower under the line of credit that stayed open, well it's unlikely that the title insurer has liability because you have personal knowledge of your line of credit and you have an obligation to make certain it is closed and you also have an obligation to not use it again.
If you are the purchaser of property and you find that a line of credit for a prior owner is still open and unsatisfied, contact your title insurer. If you have an owner policy, they have a duty to protect you and rectify the situation.
Hope that helps.