The closed interval $[a,b]$ has measure $0$ when $a\ge b$, so this was presumably in a context where one knows that $a<b$.
A sort of contrapositive would say that if an interval does have measure $0$, then it is not a closed interval $[a,b]$ where $a<b$. That certainly does not mean it is an open interval.
It is clearly false that if an interval does not have measure $0$, then it is open. Any interval $[a,b]$ where $a<b$ is a counterexample to that statement. Moreover, if you do show that every open interval has positive measure, that's not at all the same as showing that if some assumption holds, then an interval is open.
The subject line says "Why does an open interval NOT have measure zero?", which is a different topic from what the question asks. The answer to the question in the subject line is that the Lebesgue measure was deliberatly designed so that the measure of every interval is its length.