It is possible to keep your kitchen tools too clean. Like your skin, a certain amount of oil can be protective against the ravages of the three W's: Wind, Water, and Waste. It keeps air (wind) away from metals, preventing rust. It repels water, which can swell wooden tools and contribute to rust on metal ones. It prevents stains (waste) by sealing small openings in wood and stone, creating a better non-stick surface.
Unless you're working with cast-iron, a light coating of a stable vegetable oil after washing and drying will do the job just fine. I recommend putting a few drops on a paper towel and rubbing it into the surface. If it seems too oily, use a clean paper towel to wipe away excess. Use a heat-friendly, shelf-stable oil like canola, vegetable, safflower, or grapeseed. Olive oil will go rancid, and both olive and coconut oil will scorch and smoke with high heat.
If using nut oils, warn guests about allergens before they eat your cooking.
I have ten year old wooden spoons that see daily use and still look new. The oil keeps the wood from drying out and cracking, and creates a water-repellant surface that prevents staining. If you're using a pretty wood like olive, it brings out the beauty of the wood grain.
Stoneware cooking tools such as pizza stones develop a patina over years of use that serves as a non-stick surface. That patina can be helped along by oiling the stone between uses. Then the oil settles into the stone and polymerizes with heat, creating a seasoned layer (just as it does with cast-iron).
I have old-fashioned steel cookie sheets and muffin tins that need a little love and care. A light oil coat between washes seasons the surface and creates an airtight layer that prevents rust.
Back in my early 20s I had the ubiquitous cheap pots and pans with a painted-on layer of teflon that would start to flake and crack after a year. I found that I could extend the life of these by oiling them after washing. I believe that the oil would polymerize and create a protective layer that helped the cheap teflon to adhere better to the metal surface, but that is only a guess. The important part is that I could stretch the lifespan of cheap cookware when money was tight.