Do veneers stain?

The short answer: ceramic is far more colour-stable; composite can pick up pigment over time. The long answer has nuance.
What actually stains
Coffee, tea, red wine, some sauces and smoking. Also accumulated plaque, which holds pigment at the margins of the restoration.
The margin counts too
Even when the veneer surface resists staining, the line where it meets the tooth can darken if there is leakage or if hygiene does not reach that area well.
How to maintain them
Thorough daily hygiene, regular professional cleanings and polishing when the material needs it. With that, most colour changes are prevented or corrected in time.