I saw a hilarious TV clip about why this is wrong.
Problem:
The number of the verb does not match the number of the subject.
Explanation:
The pronoun “none” means not one.
The pronoun “one” is singular and takes the third-person, singular form of “work”, which is “works”.
Seeing this, we get “Not one of them works.”
Converting “Not one” back to “None” gives us the solution.
I believe that the problematic sentence is an example of speakers and writers being distracted by the nearness of the verb to the pronoun “them”, which is plural.
Solution:
“None of them works.”