Character mistake: When George and Elaine go to the coffee shop after The Movie without Jerry, they establish that Elaine moved to NYC in 1986. Then they make fun of the way that Jerry throws up; both imitating Jerry. The "Dinner Party" episode is after the "The Dog". In the "Dinner Party" episode, Jerry brags that he hasn't thrown up since June 29th, 1980. The black and white cookie ended this streak. Elaine would not have been around to see this, not arriving in NYC until 1986.
The Library - S3-E5
Character mistake: When Marion mentions Mr Bookman she says he'd been on the job for 25 years. Yet later on he tells Jerry he's been on the job since 1971, which in 1991 makes it 20 years.
The Letter - S3-E21
Character mistake: When Lippman is recounting his conversation with his accountant, he mentions that the accountant, a man, gave the tickets to "her daughter."
Character mistake: When George tells Jerry about his date with Anna, he says he is picking her up in his father's 1968 GTO, but the car he is leaning on in the park is a 1967 GTO, which has a completely different body style.
The Bottle Deposit (1) - S7-E21
Character mistake: When Jerry tells Kramer and Newman that they left their groceries under the hood of his car, he comments, "The AAA guy said I was this close to sucking a muffin down the carburetor." Jerry's car is a Saab 9000, which are fuel injected.
The Fix-Up - S3-E16
Character mistake: George calls Jerry to tell him about sex with Cynthia. He starts by saying he just got home, which Jerry repeats. A few lines later, George confirms they had sex at his own apartment. Why would he be getting home from his own apartment? (00:13:50)
Character mistake: When Jerry is being questioned on the lie detector test about "Melrose Place", the officer questioning him asks him two control questions that he responds to in full sentences. The questions should have been simple "yes" and "no" questions.
Character mistake: George makes a big deal about wanting to say "I love you" to the girl he's dating and says he's never said it to a girl, just a dog once, and he wants to say it just once. But in s02e01, "The Ex-Girlfriend", George tells Jerry he had to say "I love you" to Marlene when she told him she loves him.
Character mistake: Jerry is shopping with Vanessa; in the store there's a sign with a special offer for the salad bar, "À la cart", instead of "À la carte" (00:05:30)
Character mistake: George says that he's paying the chiropractor only half of his $75 bill. However at the end episode Jerry tells George that he gave the chiropractor $35. (00:12:50 - 00:19:20)
The Robbery - S1-E2
Character mistake: When Jerry is selling his couch to Elaine, he tells her to take it out of the door and leave - but since the arrangement is that she would move to his place, that does not make sense. (00:12:10)
The Robbery - S1-E2
Character mistake: At the end of the episode, the waitress introduces her next door neighbour, a hot redhead. After saying her line, she leaves, and George says "Nice to meet you..." and is stuck shaking hands with Carol, who he already met before and invited him to the party to begin with. (00:21:45)
Chosen answer: George does mention his brother in the episode "The Parking Space". They actually really never mention anything that contradicts the fact he had a brother though, it just isn't mentioned.