Can naltrexone cause liver damage?

Yes. There is a risk of liver damage, this risk is increased with preexisting liver problems or very high doses of the med. Usually your doctor will make that assessment and monitor your liver enzymes before and while you are on it. Properly monitored the risk is low.
Generally, no. as far as I know, liver problems were even removed from the package insert. It happened in high doses: 300 – 400 mg per day. The patients who had problems were also taking other medications. My answer is to the best of my knowledge. Of course, from the medical legal standpoint I have to tell you that you have to follow the information on the package insert. What are taking naltrexone for?
Rarely. It is my understanding that the elevations in liver enzymes reported with Naltrexone primarily occurred in trials done for eating disorders/obesity. In these trials the medication was being tested up to 350 mg daily, 7x the usual dose. The likelihood of any significant adverse event involving the liver at commonly prescribed doses is very, very low.