After Fry starts falling, the trash bin next to the table disappears. It's constant with the future episodes too.
The existence of Radio City Mutant Hall is inconsistent with the series describing mutants as needing a pass to even venture above ground. However, for example, the Washington Redskins is not a team composed of Native Americans, nor are the Vikings all Danes or Swedes; the building could well be called "Radio City Mutant Hall" to obtain a reflected glory in much the same way.
When in the bar, Bender says he doesn't need to drink. In later episodes it is said to fuel their power cells, although it could be argued he could just drink pure oil.
Bender bends the bars in the Criminal Room of the Head Museum. In the next scene with Bender in the foreground, the bars are back to the original position.
In the scene in Old New York, where Leela quits her job and Bender steals her ring, she clearly puts the ring back on her finger. However, when her hand reappears after being hidden behind her leg, the ring is gone.
Leela wasn't wearing a ring prior to the handholding scene with Fry.
In the beginning of the episode, the Cryogenics building stays on the Old New York surface. But afterwards, the building's on the surface of New New York.
The career chips of the previous Planet Express crew come from an envelope stating they were found in the stomach of a space wasp. However, it is learned in the future episode, "The Sting", that they were killed by space bees.
No one reacts oddly or in alarm to Fry saying that Leonard Nimoy was Spock in public like they did in the future episode, "Where No Fan Has Gone Before".
While Fry is frozen, aliens come and destroy civilization twice, but there are no ruins for the medieval civilization that rose up after the first time.
During the complete Earth shot in the countdown scene, the western hemisphere was shown in complete daylight, which does not correspond with Fry being in New York City at midnight.
The career chips system introduced here, in this episode, are contradicted several times throughout the series through job changes, incompetents, examples of a free market economy and references to the "Earthican dream".
Professor Farnsworth's slippers change color and shape from one scene to the next when in the Planet Express building.
The man in Leela's poster, who gives a thumbs-up, has five fingers on each hand. People within the Futurama universe, maybe apart from select mutants, have four.
When Bender snapped the bars in the Hall of Criminals, there were 2 bars left on each side of the window, but a quick close-up showed 3 bars left on each side, and then the scene resumed with the 2 bars.
When Fry and Bender are at the bar, the barman hands Bender a bottle of liquor. He takes a big swig of it, but the level in the bottle remains the same.
When Fry falls into the cryogenic chamber his party horn drops right next to the chair. When it cuts, it's gone.
When Fry is in line for the booth, he opens the piece of paper showing the picture and writing. Just before he steps into the booth, it changed color and went blank, then in the next shot it's back as before.
In the countdown scene, it shows various countries counting down from 10, but all the countries would be in different time zones.
The last set of buildings built before Fry wakes up looks totally different than it does when Fry wakes up and looks out the window.
At the Cryogenic facility, right where Fry shouts "Pizza delivery for..." the desk is positioned almost parallel to the window, but when Fry sits down at the desk, it's positioned perpendicular to the window. And Fry can't have moved the desk because in both shots the freezing chamber is placed behind it so he would have had to move that as well.
When Leela is calling for backup outside the Head Museum, her arm band switches from her right arm to her left.
When Fry and Bender are in the museum and locked in the "Villains" room, Bender pulls out the two bars in the window, and then his arms fall off. As he puts his arms back in, you can see behind him that the bars are back in place in the window.
David Duchovny is seen in the Head Museum, but Calculon claims to be him in "The Honking."
Just before midnight on December 31,1999, Fry says to himself "Here's to another lousy millennium". Although that is a year too soon, it is a common mistake and Fry is not very sophisticated to begin with. So he shouldn't be expected to know the difference.