There's only these few main things to attend to:
Use proper compressor oil, not ordinary motor oil. The reason is that compressors have a high "compression ratio" compared to a car engine, so they will combust an ordinary motor oil. Compressor oil is some sort of low residue low combustability oil. That is, it doesn't burn easily and when it does, it doesn't leave much behind.
No need to overdo it with oil spec. So long as it is compressor oil, for ordinary workshop compressors we are not talking about very high-stressed machines. So no need for expensive silicone oils or anything like that, in general.
A compressor uses very little oil, but do keep an eye on the oil level.
Don't let the cooling fins of your compressor become clogged. Mixtures of oil and wood dust/shaving would be bad for keeping heat in. Carefully using a bit of degreasant might be a good thing. Keeping the fins of your compressor's cylinders really clean can only advantage the compressor, and you will get at least some advantage because the cooler the compressor is the more air it will pump. Probably not by much, but well, it's nice to know that when you are happily working away, there is probably good reason and you have deserved it by keeping your compressor happy.
(Richard Smith, June 2005)