Continuity mistake: When Eduardo finds the letter on the mantlepiece about them apparently stealing the idea for Facebook and asks Mark if they did or not, he places a bottle on the mantlepiece just before he finds the letter. This bottle rotates slightly between shots without him touching it after placing it down.
Factual error: When Mark types email addresses in to tell people about facemash, he writes to several people @harvard.edu. At the time the movie takes place, undergraduate email addresses were all of the form email@example.com. Furthermore, the network brought down by facemash would have been referred to as the FAS network. (FAS stands for Faculty of Arts and Sciences; the eponymous network covered all buildings within the College and Graduate School of Arts and Sciences.).
Continuity mistake: When Sean shows up at the Palo Alto house Mark goes to the refrigerator and tosses Sean a beer. Right after this he tosses a beer to Sharon, Sean's friend. The bottle shatters on the wall and a few pieces get stuck in between the bricks. When the angle changes all of the pieces are instantly gone. A few seconds after this, right after Sean talks to the "wired in guy" the pieces are back.
Factual error: The head of Harvard's office is described as one hundred years older than the country dating it as 1676. When the twins leave they break a doorknob and it is given back with the line "here's your 335 year old doorknob" which would place the action in 2011. In fact it's 2004/5.
Factual error: The British character Mr. Kenwright claims his daughter is "majoring in French Literature" at Cambridge University in England, but this is an American expression. English universities do not use the major/minor system; instead, students "study" or "read" their subject of specialization (e.g. "she's reading French Literature at Cambridge").
Revealing mistake: When Mark gets a nasty note from a Harvard classmate and storms out of a computer class, a directory sign saying "USC College of Letters, Arts & Sciences" (ie. University of Southern California, not Harvard) can easily be seen on the hallway wall through the door as he opens it and walks out.