Question: How do we know that dark matter is dark, in the sense that it doesn't give out any light or absorb any? It is impossible for humans to be watching every single wavelength. For example, what about wavelengths that are too big to detect on Earth?

Answer: 
If dark matter emitted very long wave lengths of electromagnetic radiation it would mean it is composed of charged particles. There is no escape from that conclusion. Somebody might propose that dark matter is some strange configuration of charged particles which acts as a very long wavelength antenna. That might be a good model, but there is a hitch with it. If it emits longer wavelength radiation it must be colder. The Wien displacement law is that $\lambda~=~c/T$, for $c$ a constant, which may be calculated, but is not relevant here. For $T~=~2.7$K the wavelength of radiation is $.1cm$. Let me assume that there does exist some very long wavelength radiation, say $100m$ as the lower bound on this. The ratio of temperatures gives that
$$
T\_{100m}~=~\frac{10^{-3}}{10^2}2.7K~=~2.7\times 10^{-5}K 
$$
So all this can be is a very cold gas, which is the hitch. This gas is much colder than the background radiation and not in equilibrium. So from some physical grounds this is not likely, and dark matter is most likely not some ordinary form of matter that interacts by EM.
Aside from the very good theoretical reasons dbrane gives, I'd say that we don't have observational evidence that DM is truly dark. Some of the largely discredited MACHO candidates, like large numbers of red dwarfs, brown dwarfs, planetissimals etc. have been rejected for theoretical -not observational reasons. DM, that is lightly emitting is not currently detectable.