IANAD, but as a WAG, I’d say…
A first degree burn (similar to a sunburn) might result in surface peeling but will not affect a tattoo.
A second degree burn will cause blistering. If it’s extensive enough, or bad enough, there might scarring that will distort or mar the tattoo.
With a third degree burn the skin is burned full thickness - including the tattoo. The portion of a tattoo suffering this level of damage would be gone. I suppose, after skin grafting, you could get the tattoo “repaired” but grafted skin is different in texture than normal skin which may complicate things.
If someone has extensive tattoos it may be they have to “harvest” some of the inked but intact skin for grafts, and I have no idea how that would turn out.