Little! And only on the chain (not on the cogs etc.). At least that's what my mechanics' prof said.

Basically: after cleaning the chain, rinsing with water and patting dry (at that point there still are some black marks on my rag so I guess I'd have to take it out to clean it really thoroughly), then a very thin trickle of oil on the whole chain, then spin the pedals (with your hand) for say 20 turns. Let dry and wipe the next morning. In my case the bike stays in my appartment (no garage) so I wipe it before I take it back inside (I clean it on the balcony).

In any case my chain still gets darker on the next ride (especially if I ride on dirt or non-asphalt surface) and I get chainring marks anyways. But it's still better than it was before I learned how to clean it!!!