What do you call someone from the Sun?

I would use Solarian which appears to be the latin demonym for denizens of the sun as pointed out in the comments. Wiki

It also follows nicely in the footsteps of Martian, Venusian, and Jovian, which are the most common terms for the hypothetical inhabitants of Mars, Venus, and Jupiter respectively.

As for the usage in science fiction as pointed out in the comments, the examples I located were not used in the literal sense. This makes sense, as most biological processes would find the surface of the sun to be somewhat less than ideal.

Next to "Solarian," you might want to consider Helian.

HELIAN

from Greek helios (Sun) + suffix -ian

A learned synonym for the term solarian - it is derived from the higher-status Greek, not from the lower-status Latin. The term is indicative of its user's erudition. Digital Commons @ Butler University

heliacal: relating to or near the sun M-W

So what should you call someone from the earth? Tellus was the goddess of the earth, which gives us Tellurian (the preferred choice of E.E. "Doc" Smith). The Greek counterpart is Ge or Gaea, from which we get words like geology and perigee. A person living on Gaea would be a Gaean.

"Terra" (and consequently "Terran" and "terrestrial") is seen a lot in SciFi, but it's no more a proper name than "earth" is. "Terra" is simply the Latin word for "dirt" or "land." It's not the name of a god or goddess, so it doesn't follow the rule for the names of the other planets.

"Earthling" is awfully retro and "Earthian" is just beneath contempt.

Unlike the earth, the sun does have a name: Sol (with the Greek equivalent being Helios). So an inhabitant of the sun would be a Solarian or Helian. io9.com

Another possibility, by analogy with H. G. Wells' "Selenite," is Solarite.

I think Solarling or Sunling would be interesting, seeing as we are called "Earthlings". It'd be a nice name for a race a bit on the "cuddly" or more "familiar" side, though, as the name gives the connotation of that- which I doubt is what you are going for.