What would cause oil to burn if not leaking oil? Can oil leak from the oil cap? Have oil residue on top of engine

How much oil do you have to add? An oil cap can certainly leak but if its a significant amount of oil you are adding and theres only a small amount of residue by the cap there must be another issue. You can have oil leaks without really being able to see puddles where you park especially on an audi like yours where theres a plastic cover underneath the engine. If there really are no leaks you may have significant engine issues like bad piston rings. I would recommend taking it to a quality shop to have them find where your oil loss is coming from.

I'm not sure what years are involved , but Audi just redid my rings and Pistons (for free ) because my car failed the oil consumption test. Might want to check it out.