> “Burning up” pretty much is “burning down”?

Except that if you are burning up, you have a fever, but if you are burning down, you might be planning on illegally collecting some insurance money :)

Houses that are burning up are also burning down, but people who are burning up are not also burning down.

A structure burns down (to the ground). Anything else (including people) burns up.

"The house burnt up", is not proper, but people say it and everyone knows what you mean.

Then there's the saying of what you're referencing, "burning up" as in you feel like you are "on fire", when you are sick.

Phrasal verbs are probably the hardest part of being fluent in English.

People will understand what you mean, but it won't sound right.