I make your hair fall
Hair falling out, as in from radiation/chemotherapy
Your heart begins to fall, too
Radiation can cause heart failure
But I don't cut them.
Radiation doesn’t cut either hair or heart
I can be a rock
Radioactive rocks exist, since they are just rocks containing radium
In some places, I'm some dust
Radioactive dust is otherwise known as nuclear fallout
You won't live much more.
This tends to kill people reasonably quickly
Uranium is radioactive
An underground toxic find
Which can be toxic and found underground
I destroy your brain.
Radiation in high doses can destroy the brain
The hint in the title
Is related to the link in the original answer, noting that the lack of magnetosphere on Mars makes radiation exposure much, much worse there than on Earth.