It was proved by Bahouri et al. [9] that the Schrödinger equation on the Heisenberg group Hd, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on Hd for the linear Schrödinger equation, by a refined study of the Schrödinger kernel St on Hd. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on Hd derived by Gaveau [19], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results we establish local Strichartz estimates and prove that the kernel St concentrates on quantized horizontal hyperplanes of Hd.