Things that are vaguely GAN-shaped might play some role in post training. Though no one would call them GANs or identify them as the key underlying technology.
There is RLHF with a model that's trained to emulate a human evaluator, but the evaluator is not really trained jointly with the main model to adapt to its distribution and tell it from real text. Though I'm sure there are some niche cases when this is done. But definitely not a prominent thing.