There are really two concerns on an air compressor ... health of the person breathing the air and health of the compressor itself.
First off, to insure the quality of the air you are pumping into the tanks, take care of your filters. Even if your compressor is putting out wet oily air, your filters will scrub the contaminants from the air. However, if you don't take care of your filter chemicals and change them as needed, you will lose the quality of the air.
Then, compressor health is just that ... the condition of the machinery. These aren't complicated pieces of machinery but do require maintenance.
The three most important things you can do for your compressor are .... change the oil, change the oil and then change the oil!
Oil is the life's blood for the compressor. It takes heat away from the metal, it lubricates and it washes away wear products and gunk. Most of the manuals say to change the oil at very high hours ... some as high as 2000 hours of run time ... but I am a big fan of changing a LOT more often. It may be a waste of oil, but that is a small price compared to the price of repair parts.
Then the next thing is to just pay attention to the machine in general. If it is hot, cool it down. If it develops a rattle, it is most likely just a loose clamp somewhere. You can periodically take the appropriate wrenches to the fittings, clamps, bolts, etc. and make sure that every thing is snug and nothing has vibrated loose.
Finally ... KEEP IT CLEAN!!
An oily compressor will collect dirt and that acts like a blanket that holds in the destructive heat. By giving it a wash periodically, you will run cooler. Plus you won't be embarrassed for your customers to see your compressor room.
Hope that helps.