We expect heat to flow from hot to cold, devices that deviate from this are thermodynamically unlikely, which is another way of saying that they require a low entropy source. (As you said.) Low entropy = thermodynamically unlikely. This means that heat pumps are extremely non-random. So any system that looks like its random (a hot cup of tea) is going to be a very bad candidate. Similarly I think that things like weather phenomena are a bad place to look.
Living creatures can do thermodynamically unlikely things. As an example lots of (all?) individual cells move various chemicals (like salt) against the density gradients (so they move salt from a place of low concentration to a place of high salt concentration). This is Active Transport. This is just as thermodynamically unlikely as a heat pump, but its a "salt pump" not a "heat pump" so its not exactly right.
My feeling is that an actual "heat pump" (with heat, not salt) must occur in some organisms, and I think I have found a borderline example at this link (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3962001/#:~:text=After%20getting%20hot%20enough%20the,the%20nest%20surface%2015%2C%2046.) :
"In the spring ants are observed to create clusters on the mound surface as they bask in the sun. Their bodies contain a substantial amount of water which has high thermal capacity making ant bodies an ideal medium for heat transfer. After getting hot enough the ants move inside the nest where the accumulated heat is released."
If we suppose that, as a result of this, the inside of the ant's nest ends up warmer than the air outside then this I think possibly counts. Its a heat pump where the working fluid is living ants. The cold ones leave to bask in the sun, then return hot.
Its borderline because there is cheating going on, in that the sun is much hotter than the inside of the ant's nest (I assume), and they are using the sun to heat themselves up. Ideally we need ants that carry around little compressible air sacks they can inflate inside and deflate outside, so that they can unambiguously take heat from the cool air outside to deposit in the hot air inside their nest.
I have wondered something very similar to this myself. I think (at least in most cases) it is easier, on evolutionary timescales, to adapt to local climate conditions, rather than develop the machinery (and spend the metabolic energy) fighting against those conditions.
As far as I know, there are also no organisms that directly extract metabolic energy from wind, wave, tidal, other mechanical motion. Chemosynthesis based on thermal gradients AFAIK only happens in bacteria near hydrothermal vents. I assume any biological heat pumps that could exist would need to be macroscopic to be useful, but really insulation, coloring, and evaporation are just simpler.
Warm-blooded creatures (mammals and some birds) move heat around their bodies fairly effectively.
Deep-water heat sources set up convection currents that counter the "natural" gradient.
Warm blooded creatures use tricks like evaporative cooling and flow control to move heat around. Some birds, for example, have the veins and arteries in their legs tangled up in such a way that the blood headed toward their feet can give heat back to the cooled blood that's headed back toward the heart. This prevents a lot of heat loss, but doesn't actually move heat from low- to high-density regions within the body. Most of the heat involved comes from chemical processes within the cells releasing energy that, ultimately, came from sunlight.
Convection cur
An example I encountered when trying to understand jet streams is the Ferrel Cells. Hadley Cells and Polar Cells in the atmosphere are simply heat engines driven by air being heated and rising and cool air falling.
The Ferrel Cell is driven by air dragged along by the Hadley and Polar Cells, this means air dragged downward at 30° Latitude by the Hadley Cell is compressed, warming surroundings and moving along the ground to 60° where the Polar Cell drags the air up, expanding it allowing it to absorb heat from cool air around it before returning to 30°. Apparently consuming ~275 terawatts with a COP of 12.1
https://en.wikipedia.org/wiki/Atmospheric_circulation
When a whale dives after having taken a breath at the surface, it will experience higher pressure, and as a consequence the air in its lungs will be compressed and should get a little warmer. This warmth will diffuse to the rest of the whale and the whale's surroundings over time, and then when they go up to the surface again the air in their lungs would get cooler. I suppose this isn't really a continuous pump, more of a single action which involves pressure and temperature.
Any animal which is capable of altering it's own internal pressure for an extended period of time should technically qualify, since pressurising an internal cavity will make the gas or liquid within hotter (and this heat will eventually radiate to the animal's surroundings). Then the animal can cool down by reducing it's internal pressure. This effect might be negligible for the low pressure differences produced by most animals, but should still be present.
Bivalves use their powerful bodies to suction themselves to a surface, and sea cucumbers can change their internal pressure to become rigid or flexible. You might have some luck there?
Theoretically, humans should be able to do a very small amount of heat-pumping, by taking a large breath of air and then compressing it as much as possible using your diaphragm and chest muscles. This should cause the air to heat up a little (though I doubt it would be noticeable).
I don't think of fire as moving heat around so much as converting chemical energy (derived from electromagnetic) into thermal.
It creates a local area that is higher temperature than it would be without the photosynthesis->chemical energy->fire chain of events. Since energy is conserved there must also be a local area that is lower temperature than it would otherwise be, and I can't think of a way it would be cooling something hotter than the fire.
It's a weak example, but the direction of search I have is "Find a process that results in higher local temperatures than would exist without that process". The other competitor is planet and star formation, which (oversimplified) took a gas cloud very slightly warmer than the cosmic background radiation and concentrated much of it into hotter stars and planets. My engineering thermodynamics education breaks down in astronomical space, because it doesn't actually enumerate all the assumptions it makes, including "acceleration due to gravity is constant over time".
Off the top of my head: I know some chemicals (like sodium acetate, found in reusable hand warmers) change form when heated and can be easily coerced to change back, releasing the stored heat in the process. I'd be surprised to learn that there aren't any natural processes that take advantage of behavior like that, but I don't think I actually know of any.
I wonder if I'm understanding this correctly: is something like sweating an example of refrigeration, since it keeps the low temperature thing from heating up? And heat pumps are different, they keep a hot thing from cooling down, but otherwise the underlying thermodynamic processes are similar?
If I've got that straight, is any evaporative cooling an example of refrigeration, but the question here is specifically wondering about heat pumps not refrigerators?
Sweating is an example of evaporative cooling, but the fancy part of refrigeration and heat pumps is the compressor, which does work on the coolant that results in the coolant moving heat from a colder part of the loop to a warmer part of the loop.
Sweating takes heat out of the skin, but in nature the water vapor then has to move all the way to somewhere cooler than body temperature before it will condense back into rain; if you follow the water cycle it's moving heat from a hotter location to a cooler location.
I think the spirit of the original question was "is there a natural system that moves heat from a cool part of the world to a warmer part of the world?"
For what it's worth, https://en.wikipedia.org/wiki/Evaporative_cooler takes the perspective (in one paragraph) that “Vapor-compression refrigeration uses evaporative cooling, but the evaporated vapor is within a sealed system, and is then compressed ready to evaporate again, using energy to do so.” So, in this perspective, evaporative cooling is a part of the system and forced recirculation (requiring the energy source mentioned in the question) is another.
heat pumps not refrigerators
Note that what is colloquially called a heat pump is the same fundamental thing as a refrigerator — equipment is referred to as a “heat pump” when it is used for heating rather than, or in addition to, cooling, but the processes and principles are the same (with the addition of a “reversing valve” so that the direction of operation may be changed, when both heating and cooling are wanted).
By that I mean are there any natural phenomena that use a low entropy energy source to move heat against a temperature gradient. I know that there are a lot of things that can be interpreted as heat engines, like hurricanes, but I wanted to know if the opposite also happens in nature.