Am I legally responsible for paying labor costs for a repair to a car that still doesn’t run?

Unfortunately, there is no simple answer. It depends on--

1) The terms under which the work was done--did they guaranty it would run (so no payment if doesn't run) or not?

2) If there was no guaranty, then the issue becomes did they do the work to currently accepted professional standards--e.g. spend a reasonable amount time doing what any competent shop or mechanic would do--but despite their best efforts, there are more problems (or bigger, or more intractable) problems than anyone had thought, and the car can't be made to run? If this is the case, they would be entitled to labor costs, since they legitimately spent time working on the car and it's not their fault it did not work. Or if you believe that they did not do a professional job and the problem is they were careless, unqualified, didn't actually do the work they said they did, or anything like that, they would not be entitled to the money.

You also have to weigh that even if you think they are not entitled to pay, if they disagree and sue you for the money, whether it is worth having to defend a lawsuit; even if you win (and winning is never guaranteed), you'll still spend time and possibly money on the defense.