Superman is invulnerable to many things on Earth, and there's the whole red/yellow sun thing. While on Earth, Superman has heat vision and x-ray vision.
Is he vulnerable to his own heat vision? Could he burn or injure himself with heat vision?
And can he see into himself with his own x-ray vision?
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.
I'd like to say that it depends on the writing. On some occasions, he's depicted as being unaffected by his own powers, and on some others he definitely is.
I accidentally came across this picture from Superman issue #139. Apparently, he was exposed to red kryptonite, because of which his hair, beard and fingernails started growing, even though they're not supposed to grow in Earth's atmosphere according to him.
Since his powers alone weren't strong enough, he had to ask Supergirl and Krypto for help. With their powers combined, they were able to help him.
However, as the other answers pointed out, it was also shown in some other medias that Superman can be affected by powers like the ones he has, or even his own powers. Take this picture for example, where you can see him shaving his barely grown beard in front of a mirror, from Superman: the animated series.
AFAIR, he used his heat vision to shave. So he can burn off his own hair without damaging his skin.
