Byrne's Superman was contradictory in that regard...
In The Dark Knight Returns, a nuclear weapon hurt him very badly, just as in other Byrne era stories. However, he was still able to fly near or through the Sun.
I guess they got around this by saying that he would be immune to solar reactions, since he was able to store the energy from them...
Uh when did the Byrne Superman fly near the Sun? He need an air supply just to fly to the moon. Now after Byrne left the power level started going up again but they so far have not reached the insane level of Silver age