Continuity mistake: At times Ricky wears his gold wedding band, and other times - especially at the Tropicana - it is missing from his finger.
Suggested correction: So he took it off while performing. I googled images of Desi and Lucy and you can see he wore a wedding band in real life. Therefore, taking it off while filming would have been a deliberate choice. It's only a mistake if the band appears and disappears within a scene.
The Adagio - S1-E12
Visible crew/equipment: Crew member shadow is visible at the foot of Ricky's bed near the end.
The Benefit - S1-E13
Revealing mistake: There's suddenly an upright piano in the living room where Ricky's desk used to be.
Suggested correction: There's nothing sudden about it. From the very start of this episode, the card table is at the center of the living room, with the couch pushed back toward the wall, and the desk is behind the couch (holding the telephone, etc), up against the wall where the piano usually is in other episodes. As the opening scene continues, Ethel asks Ricky about the sheet music on the piano, and he explains that he's been working on a musical number for the show at the Tropicana. So, for this episode's ongoing musical storyline, the positions of the piano and desk have been switched. The rearranging of furniture between episodes occurs often, and it's not a revealing mistake or a continuity error.
The Adagio - S1-E12
Character mistake: Lucy turns in for the night in full makeup and lipstick, instead of washing up.
Hollywood Anniversary - S4-E24
Visible crew/equipment: After Lucy rushes out with the newspaper to tell Ethel about all the movie stars coming to the party, when Lucy returns to the suite and closes the door, we can see the arm of a crew member and a board with "DPI" (Desilu Productions Inc) on it, at the bottom right side of the screen, and a bit of equipment at the top right corner. (00:15:00)
First Stop - S4-E14
Revealing mistake: When the bed Lucy and Ricky are in moves as a train goes by look real close and you can see the wheels on it to help move it across the set piece.
Suggested correction: Not necessarily an error. My bed frame is on wheels.
Other mistake: When Ricky comes home at 3:00 am, the toggle light switch is in the down position. Ricky places his hand over the switch plate and the room goes dark, but the toggle light switch is still in the same down position.
Character mistake: When Lucy and Harpo are doing the mirror routine, Harpo grabs the horn on his belt with his right hand. For Lucy to create a "reflection", she would need to use her left hand but she uses her right hand.
Revealing mistake: When Fred and Ethel come with parts of their apartment's ceiling on them look closely at Ethel. The piece on her left shoulder falls off revealing what looks like a velcro pad to keep it in place.
Continuity mistake: Here it's said by the Ricardos that Fred and Ethel have been married for 25 years. Problem is in Season 1's episode "The Girls Want To Go To the Nightclub" the Mertzes both say they've been married for 18 years.
Plot hole: Despite the plot, Lucy couldn't be fooled by Jimmy and Timmy Hudson. True, they could be fraternal twins but not even Lucy could mistake them for being identical as one's got a higher voice and the other's shorter.
Continuity mistake: The Mertz' check out of the hotel and are in the garage on a motorcycle. Later they are back in their hotel room with Fred napping in a chair with a hurt shoulder.
Suggested correction: Ethel and Fred checked out of room 317, intending to leave. Later, after their motorcycle crashed, Ethel and Fred checked back in and were given room 372 (time codes 00:09:05 and 00:12:15), so Fred (with his hurt shoulder) is napping in the chair in his and Ethel's new room. There's no continuity mistake.