I was assuming a wormhole-like device with a timelike separation between the entrance and exit. The computer takes a problem statement and an ordering over the solution space, then receives a proposed solution from the time machine. It checks the solution for validity, and if valid sends the same solution into the time machine. If not valid, it sends the lexically following solution back. The computer experiences no additional time in relation to the operator and the rest of the universe, and the only thing that goes through the time machine is a bit string equal to the answer (plus whatever photons or other physical representation is required to store that information).

In other words, exactly the protocol Harry uses in HPMoR.

Is there some reason this protocol is invalid? If so, I don't believe I've seen it discussed in the literature.

Because thermodynamics and Shannon entropy are equivalent, all computationally reversible processes are thermodynamically reversible as well, at least in principle. Thus, you only need to "consume" power when doing a destructive update (i.e., overwriting memory locations) - and the minimum amount of energy necessary to do this per-bit is known, just like the maximum efficiency of a heat engine is known.

Of course, for a closed timelike loop, the entire process has to return to its start state, which means there is theoretically zero net energy loss (otherwise the loop wouldn't be stable).

Can't you just receive a packet of data from the future, verify it, then send it back into the past? Wouldn't that avoid having an eternal computer?