On a billion-year-old Earth where all life is extinct, it is extremely unlikely that there would be sufficient free oxygen in the atmosphere to breathe due to the Sun's expansion.
In the year 50,000,000, the breast size of one of the women changes between shots.
The ruins of the Planet Express in the year 10,000 appear to be the one of the normal Planet Express, despite Leela making it more productive and bigger. Although this could be because production went down when Leela read Fry's greeting card, causing her to neglect ownership of the company or they moved to a new location at some point after that.
The Professor said that a new universe is always identical to the old universe, but when Leela is sitting at a table at a reserved seat on Cavern on the Green, the waiter is serving meatloaf instead of telling the history of Cavern on the Green just like the old universe.
He could, however, have meant the formation of the universe and placement of the planets, etc, as opposed to identical in every single way.
The Professor also makes changes in both succeeding universes; first, he kills Hitler, then in the next universe he misses Hitler and hits Eleanor Roosevelt by mistake. It is also worth noting that the time machine "stays" in its original location, so the Professor would be aiming and firing his weapon all the way to Germany to kill Hitler. (Since Mrs. Roosevelt was a world traveler, there is no telling where she might have been when she was hit, but it's unlikely she would have been in Nazi Germany.)
During the time travel song, when the group is in the flooded world, they are seen using Bender's legs to move several feet but the machine continued to be in the same place it was in before.
