Treat wood veneered furniture with far more care than you would solid furniture. Veneers are very thin, and shouldn't be sanded down like solid wood furniture. A ring left by a cup or glass could be a permanent mark on veneer. Because it is thin, be especially careful when moving the furniture. Veneer can chip, and once chipped it's difficult or impossible to repair. Don't set heavy objects on veneer that could scratch or score the surface.
Most wood changes color slightly over time. This is also true of wood veneers. If your furniture is in a place that gets a lot of sun, rotate the furniture every few months to ensure even coloration.
Clean veneer as you would solid wood. Use wax-free cleaners and polishes to avoid causing deposits on the veneer that may eventually require sanding to remove. Lemon oil or Murphy's oil soap are very good for cleaning veneers.
Veneers can sometimes separate from the substrate. Simply place a small amount of glue under the veneer that has separated and clamp it solidly until the glue has cured. Be sure to talk with a hardware store specialist about the type of veneer, the type of substrate, and the size of the area that has separated to find the best repair procedure.