Other mistake: Near the beginning of the movie, the Dude is shopping at Ralph's. He is writing a check, and dates it September 11th, 1991. At the same time he looks up to see a TV with President Bush saying "This aggression (against Kuwait) will not stand." The incident against Kuwait started in the summer/fall of 1990, not 1991. The Stranger (Sam Elliot) and Walter also mention the Gulf War in the present tense.
Continuity mistake: When the Dude's driving home from the doctor's, he passes a yellow truck, visible through the side window, when he throws his joint against the door window. If you compare the shot where he grabs for his joint and the next one where he sprinkles the beer, you'll see he passes the truck twice. (01:05:35)
Revealing mistake: In the scene near the beginning it shows a big guy converting a 7-10 split and celebrating. If you watch it closely you will realise that the 7 pin is lined up too far to the left and the ball is already in the gutter when it hits the pin. The movie makers could have at least bothered to repaint the pin placement lane markings so it would not be so obvious that the pin had been moved.
Continuity mistake: When Lebowski is driving home from the doctors office, the back window of his car is so covered with dirt that you cannot see out of it, yet when he looks in his rearview mirror there is a very clear reflection of the car following him. (01:05:00 - 01:16:00)
Continuity mistake: In the scene in the limo, with the Dude drinking his white Russian, some shots show him with a milk moustache, then the shot changes, and then when it returns he doesn't have a moustache. The next time the camera left the dude and returned, he has the moustache back.
Continuity mistake: When Dude is talking to the stranger at the bar the straw in his glass changes sides of the glass very often, but Dude isn't drinking in this scene.
Continuity mistake: When Donny bowls the last time, the time when he only hits nine, just as he bowls a man with a blue bag walks behind him, then it cuts to the skittles and then back to him, and the man keeps walking from where we last saw him as if he paused for the few seconds while the skittles were showing.
Visible crew/equipment: Towards the end, when the Dude starts getting angry after Walter accidentally scattered the ashes all over him, you can see a black wire under the Dude's shirt going into his pants, most surely a microphone.
Continuity mistake: In the bowling alley, when Walter says "Am I wrong, am I wrong", there is a shoe on his left, then camera switches to the Dude, and when it comes back to Walter the shoe disappears.
Character mistake: When Walter brings Cynthia's dog in the carrier to the bowling alley he refers to it as a Pomeranian while it is actually a terrier. Somewhat related, when the the Nihilists confront Dude in the bathtub, the dude refers to the ferret as a marmot. Walter later refers to the ferret as an amphibious rodent. Perhaps he was thinking of a mink.