The Errand Boy (1961)
The Jerry Lewis talk with a doll in the end.
Tom: "Let's go rent 'The Errand Boy.'"
Jimmy mentions this movie is also included in the 50th anniversary boxset of "The Nutty Professor"; also Jimmy and Jerry Lewis talk about the scene where Jerry Lewis' character is lip-synching to bigband music
Movie scene from this Movie can be seen here
Clip of Jerry Lewis' character lip-synching to bigband music
In a deleted scene from his episode, Peter imitates Jerry Lewis in one scene
In a deleted scene from the episode "Business Guy", shown in this episode, Peter imitates Jerry Lewis in one scene