That would be a first. And on top of that, I would have no idea how that would work (when you look at how an alternator works). However, a weak alternator won't drain the battery, however it may not supply the amps and voltage needed to charge it. And at the same time, a weak battery over a period of time can weaken the alternator as it will be under more load more frequently and wear prematurely.
However, can an alternator drain the battery? I doubt it as the alternator doesn't really draw power, it produces it. It is something else in the car that is drawing power and/or your battery and/or alternator is becoming weak and will be in need of servicing