This is conjectured to hold for varieties with trivial canonical bundle (Bondal-Orlov), and is known to be true for Calabi-Yau 3-folds (Bridgeland established this in his seminal paper https://arxiv.org/abs/math/0009053). In higher dimensions this is one of the basic problems in the field, and, to the best of my knowledge, very little progress has been made towards its solution.
Let me relate it to another question. A folklore conjecture (sometimes attributed to Kontsevich) is that derived invariant varieties have the same Hodge numbers (this is proven in dimension at most 3 by Popa and Schnell, an independent proof was given by Abuaf). However, birational varieties need not have the same Hodge numbers; the easiest counterexample is $X=\mathbf{P}^2$ and $Y=\mathbf{P}^1\times\mathbf{P}^1$. These are birational but cannot be derived equivalent (by Bondal-Orlov they would have to be isomorphic!).