To care for the tree slices:
Food oil finish: Wash with warm soapy sponge and dry immediately, and lay flat to dry completely. Occasionally, apply a butcher block oil, or mineral oil to keep the finish bright and the supple.
Polyurethane finish: Occasional wiping with a Murphy oil soap will keep the finish clean and supple. After years of enjoyment, they can be wiped with a natural oil (Deft, Tung, or natural colored oil base stain). This will add protection and revitalize the color of the piece.
Any cupping that occurs should self correct when the piece dries, or cools down (in the case of a trivet). Heat can be applied to the convex side if any cupping remains. The polyurethane finish is rated to 425 degrees, so it will withstand most kitchen/oven uses. Perhaps a glowing hot cast iron skillet will leave a mark, but otherwise, they withstand heat very well.
Cabinets and Furniture:
Occasional use of Murphy’s oil soap will gently remove build up from the cabinets and add oils into the finish. Do not use chemicals on the cabinets, or a scrubbie of any type, they eat away at the finish. After years and years of enjoyment: clean well, then wipe down with a barely damp cloth of Tung oil. Wipe with clean cloth till any sticky residue is removed. This will help protect the cabinets, bring back the luster, and deepen the color.