Other mistake: The scenery in the "I Love Lucy" on-stage scenes shows the Ricardos' apartment with a living room window. That was the apartment they moved into after Little Ricky was born. Before that, they lived in an apartment with no (visible) window in the living room.
Factual error: When Lucy goes to see RKO executive VP Charles Koerner after "The Big Street" was released in 1942, one of the posters on the wall in the waiting room is for "Stromboli", which was not released until 1950.
Continuity mistake: Vivian is trying on a red dress in her dressing room when Lucille comes in to talk to her. Throughout the scene as the shots switch back and forth, the dress goes from being pulled up on Vivian's right shoulder, to off the shoulder part way down her arm.