What will trigger a worst time search for a binary heap and what is the run time? I thought if the values in a max or min heap is monotonically increasing or decreasing, then this will trigger a worst case run time of $\mathcal{O}(n)$ because you will have to go through each and every single node to get to the value you want.

However, many sources state that heap has a worst case of $\mathcal{O}(n\log n)$ for (all?) transactions. 

I don't see how you can do $n\log n$ for a monotonically increasing binary heap.

Can someone clarity?
Heaps are maintained so that they are always balanced. A heap containing $n$ elements will have height $O(\log n)$. All operations on heaps take time linear in the height.