Heat and mass transport are generally closely correlated to momentum transport in shear flows. This so-called Reynolds analogy between advective heat or mass transport and momentum transport hinders efficiency improvements in engineering heat and mass transfer applications. I show through direct numerical simulations that in plane Couette and Taylor-Couette flow, rotation can strongly influence wall-to-wall passive tracer transport and make it much faster than momentum transport, clearly in violation of the Reynolds analogy. This difference between passive tracer transport, representative of heat/mass transport, and momentum transport is observed in steady flows with large counter-rotating vortices at low Reynolds numbers as well as in fully turbulent flows at higher Reynolds numbers. It is especially large near the neutral (Rayleigh's) stability limit. The rotation-induced Coriolis force strongly damps the streamwise/azimuthal velocity fluctuations when this limit is approached, while tracer fluctuations are much less affected. Accordingly, momentum transport is much more reduced than tracer transport, showing that the Coriolis force breaks the Reynolds analogy. At higher Reynolds numbers, this strong advective transport dissimilarity is accompanied by approximate limit cycle dynamics with intense low-frequency bursts of turbulence when approaching the neutral stability limit. The study demonstrates that simple body forces can cause clear dissimilarities between heat/mass and momentum transport in shear flows.