Cold dark matter is thought to fill our galactic neighborhood with a density $\rho$ of about 0.3 GeV/cm${}^3$ and with a velocity $v$ of roughly 200 to 300 km/s. (The velocity dispersion is much debated.) For a given dark matter mass $m$ and nucleon scattering cross section $\sigma$, this will lead to a constant collision rate of roughly
$r \sim \rho v \sigma / m$
for every nucleon in normal matter. The kinetic energy transferred to the nucleon (which is essentially at rest) will be roughly
$\Delta E \sim 2 v^2 \frac{M m^2}{(m+M)^2}$,
where $M \approx 1$ amu $\approx 1$ GeV/c${}^2$ is the mass of a nucleon. The limits for light ($m \ll M$) and heavy ($m \gg M$) dark matter are
$\Delta E_\mathrm{light} \sim 2 v^2 \frac{m^2}{M}$ and $\Delta E_\mathrm{heavy} \sim 2 v^2 M$.
This leads to an apparent intrinsic heat production in normal matter
$\tilde{P} \sim r \Delta E / M$,
which is measured in W/kg. The limits are
$\tilde{P}_\mathrm{light} \sim 2 \rho v^3 \sigma m / M^2$ and $\tilde{P}_\mathrm{heavy} \sim 2 \rho v^3 \sigma / m$.
What existing experiment or observation sets the upper limit on $\tilde{P}$?
(Note that $\tilde{P}$ is only sensibly defined on samples large enough to hold onto the recoiling nucleon. For tiny numbers of atoms–e.g. laser trap experiments–the chance of any of the atoms colliding with dark matter is very small, and those that do will simply leave the experiment.)
The best direct limit I could find looking around the literature comes from dilution refrigerators. The NAUTILUS collaboration (resonant-mass gravitational wave antenna) cooled a 2350 kg aluminum bar down to 0.1 K and estimated that the bar provided a load of no more than 10 $\mu$W to the refrigerator. Likewise, the (state-of-the-art?) Triton dilution refrigerators from Oxford Instruments can cool a volume of (240 mm)${}^3$ (which presumably could be filled with lead for a mass of about 150 kg) down to ~8mK. Extrapolating the cooling power curve just a bit, I estimated it handled about $10^{-7}$ W at that temperature.
In both cases, it looked like the direct limit on intrinsic heating is roughly $\tilde{P} < 10^{-9}$W/kg.
However, it looks like it's also possible to use the Earth's heat budget to set a better limit. Apparently, the Earth produces about 44 TW of power, of which about 20 TW is unexplained. Dividing this by the mass of the Earth, $6 \times 10^{24}$ kg, limits the intrinsic heating to $\tilde{P} < 3 \times 10^{-12}$W/kg.
Is this Earth-heat budget argument correct? Is there a better limit elsewhere?
To give an example, the CDMS collaboration searches for (heavy) dark matter in the range 1 to 10${}^3$ GeV/c${}^2$ with sensitivities to cross sections greater than 10${}^{-43}$ to 10${}^{-40}$ cm${}^2$ (depending on mass). A 100 GeV dark matter candidate with a cross-section of 10${}^{-43}$ cm${}^2$ would be expected to generate $\tilde{P} \sim 10^{-27}$ W/kg, which is much too small to be observed.
On the other hand, a 100 MeV dark matter particle with a cross-section of $10^{-27}$ cm${}^2$ (which, although not nearly as theoretically motivated as heavier WIMPs, is not excluded by direct detection experiments) would be expected to generate $\tilde{P} \sim 10^{-10}$ W/kg. This would have shown up in measurements of the Earth's heat production.
EDIT: So it looks like I completely neglected the effects of coherent scattering, which has the potential to change some of these numbers by 1 to 2 orders of magnitude. Once I learn more about this, I will update the question.
Best Answer
Dark matter is not the only possible source of heat in ordinary matter: cosmic rays and similar would also heat ordinary matter. Experiments searching for dark matter see a great deal of heat from cosmic rays and look very hard for but have not yet found dark matter, which is looked for primarily by the heat it deposits. This is to say: when dark matter hits a nucleus the nucleus recoils, depositing some energy in a detector, but causing very little ionization, relative to (most) cosmic rays. This energy deposition quickly (particularly in CDMS, but also in other experiments) becomes heat which (in turn) is detected directly because it heats a bolometer or indirectly because it (for example) nucleates bubbles. With careful experimental techniques that allow the energy deposited to be seen quickly and distinguished from other energy depositions. These experiments show that there is orders of magnitude more heating / deposition of energy from cosmic rays than from dark matter, and by extension this is true for all matter not well shielded from cosmic rays e.g. effectively all matter we can imagine "seeing". Actually, this is too weak a statement: even in well shielded locations (deep mines) there is much more heat deposition from cosmic rays than dark matter. So, (in effect) I think that the best recent published limit on dark matter detection will for the forseeable future be the best limit on heating from dark matter. I suppose, this assumes that we know pretty well what the relative cross section of dark matter with different kinds of matter is. I suppose that if, contrary to all expectations, dark matter interacts strongly with something not yet used in a detector and weakly with stuff that has, this could be wrong. But, that is "not expected".