Why cool? If we can make a machine that functions at the temperature of lava, we don’t need to worry about that.
Stranger, you seem to be implying that a refrigerator cannot work.
Bender is composed of 30% iron, 40% titanium, 40% lead, 40% zinc, 40% dolomite, 40% chromium, 40-50% osmium, 0.04% nickel, and 60% storage space. He was able to swim through molten lava. So it is defintely feasible.
And don’t forget being bathed in sulfuric acid. How corrosive is lava?
No, I’m not saying that heat pump or refrigeration cycles cannot work. I’m saying that you can’t pump heat incoming from a reservoir (the lava) back out to it without creating more heat inside your closed system (the lava subterran). The only way you can effectively pump heat out of the system is to prevent it from re-entering the system, i.e. have enough insulation that it can only go in one direction (out), and in order to do that you have to raise the temperature at some point in the cycle to above that of the external heat sink. This is neither theoretically possible nor physically practicable, and in fact if you had adequate insulation to prevent heat from coming in from the lava you wouldn’t need any kind of cooling system.
I deal with applied thermodynamics every day in rocket systems, and in many ways, passive thermal protection (insulation) and active cooling mechanisms (ablative cooling, regenerative cooling, et cetera) are all ways to just barely keep the temperature below the operating limit for the short duration of the flight until you can dump the stage or leave the atmosphere and heat generated by skin drag and ram pressure. And this is with heat fluxes that are tiny compared to what a vessel would see in even a fraction of a second of exposure in molten lava.
Stranger
Precisely. This is the point I made above 9and Sunspace agreed with).
I can see trying to cool if you want to have an electric motor – lava temperatures have to be above the curie point of and metal or ceramic, I think – but who says you need an electric motor?
I think the point us to use a powered heat point to remove heat from some part of the interior of the device, cooling it to a temperature below that of the external magma, and transfer that heat to the outer part of the device, which would be at a temperature higher than the external magma. The heat pump itself would of course consume power and generate heat in the process, and the device as a whole would be adding heat to the surrounding magma. It is no different in principal to what a refrigerator does (lowers the temperature of part of the interior at the expense of power and dumping heat to the surrounding environment). We just need to figure out how to make a heat pump and radiator system that operates at a temperature greater than that of magma.
“…the sperm then burrows through the outer layer of the egg…”
Who says the system is closed? You dump heat into the immediately surrounding lava, turning it into hotter lava, and then presumably the hotter lava flows away and is replaced by ambient temperature lava.
Sounds to me like it would be easy to make a machine to solidify lava by cooling it. Make a U tube of high temp alloy to submerge in the lava, run a tube of the same material out of the lava to form radiator, and then back into the device. Pump molten salt or something through it as a transfer medium. Lava cools and solidifies on the outside of the U tube.
The entire machine isn’t submerged in the lava, and this is probably not the idea the OP has in mind, but it shows that the general concept isn’t impossible.
This volcano in Tanzania might be of interest in that respect.
Also running a machine in its carbonatite magma, at a mere 1000F would be less of a challenge than running a machine at 2000F.
We already have such a machine, it’s called a lava lamp! 
Actually, I think solidifying the lava at the heat exchange would be the last thing you wan to do. You need to move the heat away, so you need low viscosity. You don’t want heat exchanging at all anywhere else, or it defeats the purpose like a poorl insulated house.
Oops, Oldoinyo Lengai volcano, Tanzania:
http://geology.com/press-release/carbonatite-lava-at-oldoinyo-lengai/
In addition to being much cooler than ordinary lava, the carbonatite flow has the viscosity of motor oil.
Good luck with that.
Stranger
Simplifying what Stranger is saying, I think - you can’t build a refrigerator if the refrigerator is powered by a heat engine that resides inside the refrigerator. Refrigerators normally depend upon an external source of energy to work. Pretty much all our mechanisms for making substantial amounts of energy are heat engines. If you have a cool module as part of your lava submarine, fine - but the power source that keeps it cool will still need to run at lava temperatures and higher.
In terms of materials science a lava capable machine is probably less extreme than a rocket motor. Especially turbopumps, they must be some of the most extreme engineering about.
As has been mentioned above, assume the entire machine runs normally at elevated temperatures. You could use hydraulic control systems - all the way up to hydraulic logic. A simple hydraulic computer may not be beyond the bounds of possibility. Not exactly going to be tiny, but compared to the logic that drove many space probes, not incapable either. Fashioning logic from lumps of tungsten and flowing a suitable molten metal through channels etched into it could yield a simple controller that would happily run a probe.
As has been noted - where things get unstuck is in powering the thing. About the only mechanism that is going to yield enough power and run long enough to do enable a probe/machine to do useful work is a nuclear powered heat engine. And that engine is going to run really hot just to generate a modicum of power. Something of the form of a Stirling engine probably. The engine will probably have to pump lava though itself to keep cool, as well as sink the waste heat.
In all it does sound rather reminiscent of the issues in building a rocket motor - just in a twisted fashion. Every thing happens much slower, but designing to make use of the conditions and problems in transiting from cold and not running, to hot and running making life for the engineers miserable.