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In response to Jim's challenge, I think I've found the easiest approach a bit belatedly: any module in category $\mathcal{O}$ is finitely generated over $U(\mathfrak{n}_-)$, and any Verma module is free of rank 1 over it. Thus, if $M$ is infinite dimensional, and $N$ a Verma module, the tensor product $M\otimes N$ is free of infinite rank over $U(\mathfrak{n}_-)$, and thus not in $\mathcal{O}$.