Grammatically, there is nothing wrong with the sentence—however, it's awkward because it can be interpreted ambiguously. To avoid the issue, the three types of tiredness need to be offset from the rest of the sentence.
Em dashes can certainly be used, but so can parentheses:
People who are tired (physically, mentally, or emotionally) need the energy that they can get from your magnetism.
It's mostly a matter of style which method you prefer.
Note that if you do want to use a colon, the sentence would need to be rephrased:
People who need the energy that they can get from your magnetism are those who get tired: physically, mentally, or emotionally.
Or, if you want to use a semicolon:
"People get tired—physically, mentally, or emotionally; they need the energy that they can get from your magnetism."
Here, I used an em dash because the first part of the sentence still had a problem with the awkward use of commas.