No, the electricity flowing will still generate heat. Flipping bits may not take a lot of power, but it does take power, and some of that power is converted into heat regardless of what material you use. Except maybe superconductors, but I'm not sure.

To amplify a bit on what ranger207 said: Normally, metals are good conductors of heat, for the same reason that they are good conductors of electricity - the flow of electrons transports the heat.

In a computer chip, the heat generation isn't (primarily) due to the flow of electricity along the conductors - it's primarily in the transistors (and IIRC, mainly in their switching).