Photography is permitted just about everywhere except power plants and military bases. Also the police don't seem to enjoy being photographed, do it when they are not looking. At Museums taking photos is extra and video is even more expensive. You can hide your camera if it is small and not pay, but don't get caught by the Museum Babushkas!!! And I recommend not trying to photographs President Residence, the secret police get really annoyed.

No cameras or bags allowed in Red Square when the Lenin Mausoleum is open, (it was closed). You will have to check your camera and bags in at the Camera Storage facility under the walkway leading into the Kremlin if there is any photo restrictions in place the day you are there.