This is a common problem. Usually, it’s because the lights are no longer serviceable. I believe most exterior Christmas lights are only intended to be used for a total of 90 days. Although, many people don’t follow the useful life guidelines and use them for years.
Although possible, I doubt the GFCI is bad, it’s just doing it’s job – which is to detect a current mismatch between the hot and the neutral. The National Electrical Code requires all exterior and garage outlets be GFCI protected due to the increased risk of electrical shock from using electrical equipment outdoors.
You could try a different GFCI protected circuit to rule out a bad GFCI. Although I wouldn’t reccommend it, you could use an extension cord to plug your lights into a non-GFCI circuit around your house.