I've been trying to find the answer to this, but to no avail. Does the Oslo pass cover any transportation between the airport and Oslo center city? Does the Copenhagen pass?
Yes, the Copenhagen pass covers the train, bus or train/Metro from the airport - it's the same train as you'd take for other local stations. The Copenhagen pass covers all bus, train and Metro transport in Copenhagen, including to Helsingør, Hillerød etc., but I think not the private train lines to the north and northwest part of Sjælland.
The train to and from the airport in Copenhagen is covered. But the Airport Express train in Oslo is not covered, nor is other transportation modes to Oslo Airport.