In my family, children have to get permission before leaving the dining table.
No correction needed. This sentence is grammatically correct.

Lang Lieu couldn't buy any special food while he was very poor.
No correction needed. This sentence is also grammatically correct.

This factory has fewer workers than that one, so it needs more.
Corrected: "Less" is used for uncountable nouns, while "fewer" is used for countable nouns (like "workers").

I don't know which one is the most ancient: Bat Trang pottery village or Bau Truc pottery village.
Corrected: "Most ancient" is the correct comparative form. We need "the" before "most" to make it sound natural.

If you don't know how to operate the machine, you should read the instruction manual.
Corrected: "How to" is needed here instead of "what to" because it is referring to the manner or way of operating the machine.