For their environment, it depends on time
The question of whether something actually does melt under heat has to do with time - how fast the heat gets in (conduction), and how much it takes (physical properties like specific heat). However, at a certain temperature and pressure, everything will melt. A body of hot mass at temperatures like the sun will always be a plasma - it simply can't deal with the extra motion to form bonds and stay together chemically.
Keeping your cool would keep things ok
At 300 degrees Fahrenheit, not too much will be destroyed. Some rather unexpected substances like wood burn quite a bit hotter, but wood accelerates its own heating when it combusts. So it would mostly be about wearing something light enough and heat-taking that would not interact with objects in the environment too much. I would think wearing something insulating, and not metal, would be better if the temperature is always hot. Otherwise, the metal or material that transfers heat quickly would cause, well, embarrassing losses of heat and probably damage to property.
... 106
but otherwise, get a disability waiver
300F is already too hot to do a lot of things, like touch people or anything with some flammability. It's probably too hot to reliably refrigerate on your body, if that's not somehow dangerous for this person (I doubt a home refrigerator could do that, unless mad-scientist <PERSON> made it). On the other hand, 3000 degrees Fahrenheit is quite balmy, and if your meta-human has temper tantrums or gets beat up a lot, and for long enough, they would melt just about anything. As a reference, sand melts at 3000 degrees Fahrenheit (into glass), and there aren't any known substances that survive above something like 5000 degrees Fahrenheit for an indefinite time. Many things will vaporize at 3000F, like Aluminum according to Wikipedia. Your metahuman would be a dangerous foe if they weren't sinking into the ocean or melting concrete fast enough.
As an edit, I guess your metahuman could have a sort of flying superhero MO and then not get anything hot except a plume of air above them. The right suit could do the trick, in theory... 561