My understanding of the microscopic "mechanism" of blackbody radiation* in a gas is as follows: interactions between fluctuating charge distributions in particles of a gas create a microscopic random electromagnetic field. The particles interact with this field and with each other directly (for example, in collisions), producing electromagnetic radiation in a continuous spectrum that is consistent with the blackbody spectrum.
This means that only gases which actually couple to the electromagnetic field are allowed to produce blackbody radiation. This is fine for almost all normal matter** in the universe, as nearly all matter is either overtly electrically charged, or is a composite particle that has charged constituents, and therefore has a fluctuating charge distribution. (In the case of a photon gas, the constituents themselves are not charged, but the blackbody radiation and the blackbody spectrum are equivalent, because they essentially are blackbody radiation.) This is not the case for neutrinos. They are long-lived, pointlike particles with no electrical charge, and as such, they should not be able to interact with the electromagnetic field. So a gas of neutrinos would not produce blackbody radiation.
Is this believed to be true? The only "neutrino gas" in thermal equilibrium that I'm aware of is the ensemble of relic neutrinos created along with the CMB, and the difficulty of measurement of low-energy neutrinos means that we probably won't be measuring this any time soon. Nevertheless, are there observations that support this line of reasoning?
*I realize that blackbody radiation is a statistical, macroscopic phenomenon, so any semblance of a microscopic "mechanism" is stretching the truth a bit. This is why I don't endeavor to go into specifics in this explanation.
**Except dark matter, obviously. Which, by mass, should by all rights be considered the "normal matter" in this universe, but that's beside the point.