Place your veneer counter-height table out of the way of drafts and direct sunshine. It is also a good idea to keep it away from heavy foot traffic. People don't expect a table to be that high, and are therefore more likely to bump their elbows on the corners.
Clean your veneer table by dusting it one to two times a week with a soft, dry cloth, no matter what kind of veneer it is. Fine grit can scratch the surface, and the more scratches, the more danger that liquids can seep through to the wood below.
Treat spills by wiping them up immediately with paper towels. The worst thing to happen to veneer furniture is for liquids to seep underneath the veneer and soften the adhesive, which can cause the veneer to lift. This is less of a problem on expensive wood veneers, but it can happen.
Clean dried spills by adding two to three squirts of mild dish soap to a bowl of warm water. Dip a clean cloth in it and wring it out completely. Rub the dried spill until it comes off, wipe down the rest of the table and then dry it with a clean cloth.
Polish tables that have real wood veneers the same way you would polish wood. Not all veneers are made of fake materials; sometimes a thin layer of expensive wood like maple or cherry is put on over less expensive wood to keep costs down, or for reasons of strength and durability. Use a soft, clean cloth and a spray polish or oil soap. Always spray the cleaner onto the cloth, never the table.
Clean the legs on your veneer counter-height table every time you dust the top. The legs on veneer tables are more visible than on coffee tables or end tables, so check for cobwebs and drips from spills.