A little reading suggests:
It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is transcendental.
If we just want irrationality rather than transcendence, is a proof known?
Can we prove $\pi+e$ is irrational? Can we prove $\pi \times e$ is irrational?
It is not known whether $\pi + e$ is irrational, nor whether $\pi \times e$ is irrational. See $\# 22$ here.