Short Answer: No. Superman is not immune or can be selectively immune to his own powers if he needs to be. Consider him able to tweak his body's response to his powers so if he needs to view his internal organs using his "x-ray vision" or other sensory powers, he is able to do so. He can use his heat vision to shave or to burn himself if he needed to.
Longer Answer: No matter the version of Superman, (save perhaps the Golden Age Superman, whose scope of powers are vastly outdone by his later incarnations) Superman can be affected by powers similar to his own. This is evidenced by his interactions with the early Phantom Zone villains (General Zod, Faora, and Ursa) who when released under Earth's yellow sun, exhibit the same powers and resistance to injury that Kal-El does.
Superman's sensory powers are problematic but the general consensus is that he is not necessarily using x-rays to see with. The power has been described in as many ways as a wavelength modification allowing him to perceive x-ray radiation to a nearly psychic form of remote viewing. However it has been described, he has been able to see inside of other Kryptonians to check for injuries, and it is likely his power would allow him to see inside of himself as well.
I make the note of selective invulnerability because his hair is almost never burned off in a confrontation with metas who posses flame powers or when he is exposed to weapons such as fuel-air explosives or nukes. If his hair were not able to be protected by whatever form of invulnerability he is being currently written with, it would always be burned off during those conflicts.
It is also noted that earlier versions of Superman (Post Crisis, John Byrne era) as he weakened or was damaged, his invulnerability was reduced and eventually even his uniform was able to be damaged by his enemies. The battle with Doomsday is a perfect example of his powers being reduced by his diminished internal energy.