Schur measures are probability measures on Young diagrams depending on two countable sets of parameters. Introduced by Okounkov in 2003 and containing the poissonized Plancherel measure as the most known example, they lead to determinantal point processes on the one dimensional lattice. Okounkov also proved that the gap probabilities of these measures are tau functions for the 2D Toda lattice hierarchy, that is to say, they satisfy a hierarchy of particular bilinear PDEs. In this talk, I will show how to extend this result to expectations of more general multiplicative functionals. I will try to give a comprehensive exposition of the techniques of the proof which use the fermionic Fock space formalism. As an application, we recover a recent result of Cafasso-Ruzza on the finite temperature discrete Bessel process, which corresponds to a deformation of the poissonized Plancherel measure, and also obtain a hierarchy for more general finite temperature Schur measures.