Continuity mistake: After Leela wipes the egg off her face, her cloth and cup are in front of her on the table. After Fry gives his little speech, they have disappeared.
Bender Gets Made (a.k.a. Bendfellas) - S2-E17
Other mistake: The strap for Leela's eye-patch on the right side of Leela's head (from her perspective) goes just over her ear. The left side goes above her ponytail. This would prevent the two ends of the strap from meeting.
Put Your Head on My Shoulder - S2-E10
Continuity mistake: After the accident when Fry looks at his body, there is a tool kit or something similar more or less right in between the body's legs. In a few moments there is a shot from above and the kit is several feet to the right of the body without anyone touching it.
Anthology of Interest 2 - S4-E3
Continuity mistake: When Bender is strapped to the table at the beginning of the reverse fossilization process, he has sensors on his sides, ears and ankles. During the process he magically gains alligator clips on his fingers and toes and sensors on his wrists. They spontaneously appear when the Professor pulls down the lever to start the process.
Continuity mistake: When Fry and Leela discuss his career, as Fry says "But I don't like being a delivery boy", we see Leela has her hands on the table. When it cuts her hands are under the table. (00:05:45)
That's Lobstertainment! - S3-E8
Continuity mistake: The patches on Uncle Zoid's jacket sleeves appear, disappear and change from left sleeve to the right during his conversation with Zoidberg in the restaurant.
The Honking - S3-E1
Continuity mistake: When it cuts to the first shot inside the castle when everyone is sitting at the table you see Bender sitting there with a glass on his right side. When it cuts to a close-up of him the glass is gone. (00:03:50)
Anthology of Interest 2 - S4-E3
Factual error: When human-Bender is dancing in the disco with the two girls there is a fully lit disco ball which manages to cast no light beams on the walls and floor, as they normally do.
Continuity mistake: When Bender is brought into the surgery tent and Zap is holding Nixon, the level of the fluid in Nixon's jar changes between angles. In the wider shots it covers his whole head, in the closer ones the top of his head is above the fluid.
Continuity mistake: After the fight with Destructor when Bender lies flatten to the ground, the wrestling manager appears with nothing in his hands. When it cuts he is suddenly holding a cigar.
Continuity mistake: At the end of the episode, when Bender and the Countess Delaroca are hanging on to the escape pod, the diamond on the diamond bracelet the Countess was hanging onto changes position between shots.
Deliberate mistake: The fossilised dog looks like the dog did when he was young; it should look old and decrepit, as the dog was when he died. (Presumably deliberate, to avoid a spoiler.).
The Honking - S3-E1
Continuity mistake: When we see Bender welded to the wall the paintings on his sides are both hanging about a feet away from his arms. When he gets off the wall it's obvious the paintings have moved much closer to him. (00:11:55)
Continuity mistake: The lava is close to the machine, until Fry runs to it, then it's a long way away.
Kif Gets Knocked Up a Notch - S5-E5
Continuity mistake: Zoidberg wasn't in the machine to begin with, yet he appeared later. When it spins you see the whole inside of the machine - he's definitely not there.
Suggested correction: They even make a comment about this, he lives there. So he was probably sleeping under the grate but still got ejected. There's parts of the floor we don't see.
Continuity mistake: In the intro, before the main titles, the Professor informs the crew that he's booked them on the maiden voyage of the Titanic. When Leela says "Looks nice," at the beginning of the "scary" Titanic montage, the right side of her shirt is undamaged. During the middle of the "scary" Titanic montage, it suddenly becomes torn.
Continuity mistake: When Fry and Leela dig out Santa there are cracks in the ice around him which disappear when they lift him up.
Xmas Story - S2-E8
Continuity mistake: When Bender goes off the cliff, we see in the overhead shot that there are no kids on the icy lake, nor are there any tress near the lake. When it cuts there are kids on the ice and trees surrounding it. (00:02:45)
The Honking - S3-E1
Continuity mistake: When the attorney reads up the will to the butler the attorney has no paper under his arm. When it cuts papers appear under his arm. (00:03:05)
Put Your Head on My Shoulder - S2-E10
Continuity mistake: When the tow truck appears to tow Amy's car back to Earth, the window is completely steamed. The tow truck driver only wipes away a small bit of the steam but in the next scene the window is completely clean.