Scientists have come up with material that has the properties of a thermal diode (that is, it allows heat to pass in one direction, but not the other). At first glance, this seems to violate at least one of the laws of thermodynamics, but I'll have to read more and think about it more to have a firm opinion.

The material merely conducts better one way than the other, and doesn't flow "uphill". Similar devices have been built for many years; a heatpipe without an internal wick is a good example.
Since the liquid pools at the bottom end, it can conduct heat upward very well by vaporization and condensation, but downward only by conduction in the wall. 1000:1 ratios are not unheard of.

Hmm. Seems no less physically possible (in the sense of not violating thermo laws) to realize
than a vortex tube or a Peltier junction.

Ultimately it's just another heat pump.

Key words from the article: "Strung together in the right way, some biological molecules, such as DNA strands, might have the required properties, the researchers speculate. Living cells may even control the flow of heat energy this way already."

Living cells use energy to do this, from metabolism, ie: chemical source of energy.

Any real device either would have to use real Maxwell's demons (as yet not physically available), or suck energy from somewhere.

Just how to supply the energy required is the question, and that depends on the details of the design.

Quinbus Flestrin