The noun “needful” means something needed or requisite, while the noun “necessary” means an indispensable or essential item, so they are actually synonymous. The more current word, though, is “necessary.” Personally, while saying “Please do the needful” is beyond reproach grammatically, it sounds quaint—almost archaic—to my ear. I would think that “Please do the necessary” is the more natural and idiomatic way of expressing that same idea.
