As with all motivic L-functions, irrationality of special values breaks into two rather different problems:

**Critical values**. In the case of Dedekind zeta functions, those are the even positive integers, and the strategy is to show that $\zeta_K(-2n-1)$ is rational, so that from the functional equation follows that $\zeta_K(2n)$ is irrational. Of course, this only works for totally real number fields, since that's the only case where the functional equation doesn't vanish at the odd integers.

This is precisely the Siegel-Klingen theorem, $\zeta_K(2n)$ is a rational multiple of $\pi^{2n[K:\mathbb{Q}]}$, and therefore irrational.

The actual value of that algebraic number is in general not known, see the Lichtenbaum conjectures.

**Non-critical values**. Odd positive integers. This is the (even more) difficult part, because you have to actually prove that something non-trivial is irrational. As you mention, some results are known for Riemann's zeta, but as far as I know, there are no known results for number fields other than $\mathbb{Q}$.

(On a side note, a generalized "infinitely many irrational numbers" type result for Dirichlet L-functions was published a few years ago by Masaki Nishimoto (paper here)).

notbecause $\pi$ is irrational. $\endgroup$ – Gerry Myerson Feb 25 '15 at 22:09