The only authentic shots of Mt. Rushmore in North by Northwest (1959) are the ones looking at the monument from a distance.  The shooting and subsequent ambulance scenes were filmed in the actual monument cafeteria and surrounding grounds, but nothing that required accessing the monument itself was on-location.  

While Hitchcock did have an original permit to film on and around the mountain, rumor got out that the film involved a fight scene and a couple deaths on the face of the monument, and government officials banned the production from filming there. Mt. Rushmore had to be recreated on a Hollywood set.

Shooting the scenes on the MGM Mt. Rushmore was as simple as any other Hollywood trickery using set pieces, special effects, and clever camera work.