Rotation has a strong impact on the transport processes and flow structures in plane Couette flow (PCF) and Taylor-Couette flow (TCF). Momentum transport in rotating PCF and TCF has been extensively investigated. However, rotation also affects the transport of scalars and heat in Couette flows, but this has been much less explored. Here, we present a study of the influence of rotation on both the heat and momentum transfer in PCF and TCF using DNS. We observe that a small to moderate anticyclonic rotation rate leads to significant increase in both momentum and heat transport. However, anticyclonic rotation enhances heat transport more than momentum transport. As a results, the Reynolds analogy between heat and momentum transport breaks down in both PCF and TCF. This effect is particularly pronounced in PCF and TCF subject to fast anticyclonic rotation when the heat transport can be much faster than momentum transport.