Does the Born-Oppenheimer approximation fail for muonic molecules (i.e. molecules where one or more electrons are replaced with muons)?
Deepends on your definition of "fails". The accuracy of Born-Oppenheimer approximation is determined by the smallness of the electron/nucleus mass ratio. For hydrogen this ratio is $\approx 1/1800$. Replace the electron mass by the muon mass, which is 200 times larger, and you get $\approx 1/9$ which gives you a rough estimate for the relative accuracy in case of muonic molecules.
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