Could fake/false teeth get yellow?

Yellowing of teeth. Porcelain fake teeth can develop external stain only that can be polished off. Plastic or composite fake teeth can form extrinsic as well as intrinsic stain and can yellow over time.
Some. Denture teeth can stain, unless they are porcelain. Few are porcelain anymore, for good reasons. Crowns and bridges on natural teeth or implants will not stain if they are porcelain or other ceramic material. If acrylic then they can stain. Sometimes they can be repolished and some of the stain can be removed. Tooth whiteners, pastes, etc. will not help. Good Luck.
Of course. The dentures need to be well maintained and cleaned daily or else they will start to appear yellow and stained.