For what integers $n>1$ is $\ln(n)/\pi$ irrational?
Clearly, if $\ln(m)/\pi$ and $\ln(n)/\pi$ are rational then $\log_mn$ must be rational. In particular, if there is prime dividing $m$ but not $n$, then $\log_mn$ is irrational, so one of $\ln(m)/\pi$ and $\ln(n)/\pi$ must be irrational. As an application, $\ln(p)/\pi$ must be irrational for all primes $p$ except possibly one.
It follows that either $\ln(p)/\pi$ is irrational for all primes $p$, or there is prime $p^*$ for which $\ln(p^*)/\pi$ is rational. Then for any $m>1$ which isn't divisible by $p^*$ we would get that $\ln(m)/\pi$ is irrational, and what follows, that for any $m>1$ not a power of $p^*$, $\ln(m)/\pi$ is irrational.
What can be said in general?
Gelfond's constant, $e^\pi$, is transcendental. If $\ln(n)/\pi$ were rational, say $p/q\not=0$, then $e^\pi=n^{q/p}$, which is algebraic (a $p$th root of the integer $n^q$). So no, $\ln(n)/\pi$ is never rational for $n\gt1$.