Does the hybridization of sp3d2 type exist?
First, hybridization is a model used in Bond Valence Theory to explain the bond nature between some atoms in some molecules. Actually, the current binding theory is the Molecular Orbital Theory, in which we no longer use hybridisation. But, we study hybridisation because it is a simple and useful tool to explain a lot of compounds.

But, what the professor said in your question, it is not that the sp3d2 hybridization does not exist and cannot be used to explain any molecule. What he is saying is: In hypervalent compounds, there is evidence that the d functions do not participate in the bond.
After reading your question, I have searched a lot about the binding in hypervalent compounds. And what I found was that the nature of bond in hypervalent compounds like XeOF4 has been the subject of a lot of controversy for a long time, because in the represetative atoms, the energy gap between the nsp and nd orbitals is too large, so the d orbitals cannot hold extra electrons. And, nowadays we have evidence through ab initio calculations that the d-functions do not participate in bonding in hypervalent molecules. 

However, this is a recent evidence, so, if you have to awnser in a test about the hypervalent molecules, use the d-orbitals to explain them!

I am leaving some links with some articles and books that I read about this subject:

Chemistry of Hypervalent Compounds

The three-center-four-electron (3c-4e) bond nature revisited. An atoms-in-molecules theory (AIM) and ELF study

New bonding concept for Hypervalent molecules, including electron poor and electron odd compounds