Can I filter out vanilla bean seeds?
A chinoise will not--I speak from daily experience--filter out vanilla seeds. A coffee filter would, but my question is: why on earth would you bother? It's annoying to do (and depending on what the product is, filtering through a coffee filter would ensure you lose other things), and seeing actual vanilla seeds in the final product tells your guests that you are using real vanilla beans, not extract.

Professional kitchens do not, generally speaking, attempt to remove the seeds for the reasons noted above.

@stephenmacdonald, most dessert recipes that call for a fine mesh strainer do so because there is the possibility of chunky things ending up in the product. In the case of that recipe, I would guess it is to prevent chunks of undissolved gelatine from ending up in the final dessert.