I've simplified above counterexample - there shouldn't be any doubts
now.

Everything is in vacuum, without gravity.
Take a tube with interior covered with mirror.
Fix two transparent separators inside and place hot gas between them.
Now place two mirrors on both sides, which can freely move inside the
tube.

Some of thermal infrared photons will be bounced by a mirror - giving
part of own momentum, thanks of momentum conservation law.
The heat of the gas will be slowly converted into momentum of mirrors,
which can be converted into work.

After infinite time all heat energy will be converted into kinetic
energy of mirrors - we will have only gas with T=0 and moving mirrors.

Returning to practical use/biology ...
I was thinking about 2nd law of thermodynamics and crystallization.
During this process we get higher ordering (lower entropy), but the
cost is energy difference between free and bind molecule - this energy
is usually just dispersed around, increasing general temperature.
But what if we wouldn't allow this energy to run away randomly ... for
example storing it in chemical energy of some molecule, like ATP ...

That lead me to mechanisms that could allow organisms to feed directly
with heat (not using thermal infrared):

Let say that we have two molecules(A,B) which has larger total energy
separated(E1) than when they are bind (E2 Additionally there is energy barrier between these states.
Now when they are bind in solution, their thermal energy statistically
sometimes exceed the barrier, and they split (reducing temperature!).
But to bind them back, they not only have to reach the barrier, but
they have also to find each other in the solution - it's not very
likely, so statistically concentration of AB is relatively small
comparing to concentration of separated molecules.

Now we will need a catalyst which reduce the barrier, but then use the
energy difference for example to bind ADP and phosphate.
For example it catches all required molecules and uses energy stored
in own structure to take A and B closer, to make them reach the top of
the barrier, then use energy they produce to bind ADP + P and restore
own energy.

I know - this enzyme would work in both directions, but concentration
of AB should be small, such that the wanted direction should dominate.