[breezeTrowel]

Superconductors do not generate heat when in their superconducting state. Otherwise they wouldn't be superconductors.

[Filligree]

Superconductors do not generate heat for a constant DC current. Computers are very, very AC, and you do get heat production anytime the current changes.

It is, IMO, a bit dubious whether or not anything is truly flowing in a superconductor at constant current. Electrons don't have identity, so the 'constant flow of electrons' can be rephrased as 'the physical system isn't changing'... and the degree to which you can tell that there are electrons moving about is also the degree to which the superconductor isn't truly zero-resistance.

[eig]

You can tell electrons are flowing in a superconductor by measuring the magnetic field induced around them.

[boxed]

Yea, but just keeping everything in the static superconducting state wouldn't be a computer either.