Recent work has shown that a temperature dependence of the shear viscosity to entropy ratio, η/s, influences the collective flow pattern in heavy-ion collisions in characteristic ways that can be measured by studying hadron transverse momentum spectra and their anisotropies. Here we point out that it also affects the pair momentum dependence of the Hanbury-Brown-Twiss (HBT) radii (the source size parameters extracted from two-particle intensity interferometry) and the variance of their event-by-event fluctuations. This observation establishes interferometric signatures as useful observables to complement the constraining power of single-particle spectra on the temperature dependence of η/s.