Can certain formulas cause diarrhea?

Yes. If all babies were just alike there would be only one infant formula. Yet everyone knows that babies are born with differences; this can extend into tolerance or intolerance of infant formula. Perhaps the most common are cow milk protein allergy and lactose intolerance. While not exactly the same, symptoms of increased gasiness, colicky pain, and diarrhea are common hallmarks of these disorders.
Yes, but very unusua. An infant who is truly allergic to milk (milk protein hypersensitivity) develops severe diarrhea when exposed to cow's milk protein in most infant formula.