I can't find a rigorous proof but I have a feeling it's true.
Informal argument: Suppose $a+b$ and $ab$ are rational, $a$ and $b$ are irrational (since just one can't be irrational). Then $a$ and $b$ must have irrational "parts" that "cancel out" eg. $a = \frac4\pi$, $b = \pi$, $ab = 4$, which is rational. But then $a + b$ ends up being irrational, since the denominator has the irrational part squared and the numerator does not.
Obviously, this isn't a proper proof at all, but I can't think of any examples of numbers where this argument wouldn't hold true. Can someone provide a proof or counterexample?