Plot hole: Harry tells Doc Ock that in order to find Spider-Man he must find Peter first. Doc Ock finds Peter with Mary Jane in the cafe and throws a car through the window straight at them, then later throws Peter against a brick wall. Any normal person would've been killed instantly (or very badly injured), and Doc Ock doesn't yet know that Peter is Spider-Man. Given that Peter is his only lead on Spider-Man, it makes no sense that Doc Ock would try to kill him.
Factual error: Considering the brightness of the fusion process, Dr. Octavius has to wear special goggles to be able to see it. Yet no one else in the room is wearing such goggles or seem hurt by watching the whole process, just as at the end of the movie. When welding something, no one can look at the arc that's created, as it would hurt his eyes and burn his retina; presumably, the fusion process would be brighter and more powerful than that, and so should have some kind of damaging effect on everyone's eyesight (except Spider Man's, maybe).
Continuity mistake: After Spider-Man recovers from the train incident, Doc Ock smashes in the other end of the train and throws the door on the floor in front of him. The camera cuts to another shot, and when it returns the door is no longer on the floor, nor is it there for any subsequent shot.
Factual error: Dr. Octavius says his fusion relies on tritium and that there is only 25 pounds of the substance in the world. In reality, tritium is a good deal more common than that. From this article (http://www.epa.gov/radiation/radionuclides/tritium.htm): "It is used in various self-luminescent devices, such as exit signs in buildings, aircraft dials, gauges, luminous paints, and wristwatches. Tritium is also used in life science research, and in studies investigating the safety of potential new drugs." There's also a large region of water near Antarctica which is rich in tritium.
Continuity mistake: During the final conversation between Spider-Man and Doc Ock, the rips in Spider-Man's suit keep changing. For instance, there is a tear on his right shoulder; for most of the scene, there is a single piece of black webbing left holding the rip together, but when Doc Ock grabs Spider-Man's arm, the rip now has two pieces of black webbing. Then it goes back to one.
Continuity mistake: In the scene where Peter is riding his scooter to Mary Jane's play, after the music fades out you see a black convertible being chased by several police cars. The black convertible is rammed forcibly, and you see many hideous dents on the side of the car. Thirty seconds later, after Peter is done talking to the kids, we see the same car from the same side it got rammed into, but the hideous dents have vanished.
Plot hole: At the end, when Peter saves Mary Jane from the collapsing structure, he jumps up in the air and swings her to safety. However if you look at the web he is holding onto it is hanging down vertically. What did he attach the web to that was directly above him, the sky? Every bit of the structure around him was/had collapsed and the only thing remotely close to swing from was the cranes. Yet we see that was quite a distance away, so Peter would have had to shoot the web away from himself not above him.
Continuity mistake: As Peter stops the elevated train with many webs, a tear appears in his suit on the inside left arm during a closeup. Later, when his arms are still outstreched, there are many shots where you can see that the tear has disappeared; for example, a few shots later when he is still trying to stop the train; when the passengers are lifting him over their heads into the train; and many others.