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.
You can tell there's a magnetic field, certainly. My argument is essentially one of nomenclature; I don't feel a constant electron-field should count as 'flowing'.
Of course it isn't actually constant -- there are multiple electrons, and you can tell that the electron field is quantized. But the degree to which that is visible, is the exact degree to which the superconductor nevertheless doesn't superconduct!
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.