How to prove that a binary heap can't be sorted with complexity that better than O(nlogn)
A binary heap with $n$ elements can be built in time $O(n)$. If there was a (comparison based) sorting algorithm capable $A$ of sorting the elements in a heap in time $o(n \log n)$ then you could obtain a (comparison-based) sorting algorithm $A'$ that works for arbitrary sequences $S$ of $n$ elements. Algorithm $A'$ is as follows:

Construct a heap $H$ from $S$.
Execute $A$ with input $H$.

Such an algorithm requires time $O(n) + o(n \log n) \in o(n \log n)$ and hence it contradicts the $\Omega(n \log n)$ lower bound on comparison-based sorting algorithm. Therefore, algorithm $A$ cannot exist.