1. In "The Dumpling Paradox", they start Halo night 6 minutes late, and determine if the 6 minutes should be taken out of game time, bathroom time, or pizza break. Wednesday is Halo night, so it must have been Wednesday. But pizza night is on Thursday.
2. In "The Bozeman Reaction", Sheldon allows Howard to install a giant net into his apartment in order to electrocute robbers. But, In "The Ornithophobia Diffusion", Sheldon reveals that he has a fear of nets. Since The Ornithophobia Diffusion is after "The Bozeman Reaction", Sheldon's fear of nets may have been triggered because of being trapped in a net at the end of the other episode.
3. In "The Maternal Capacitance", Penny revealed through her conversation with Leonard's mother that her father's name is Bob. However, in "The Boyfriend Complexity", when her father visits her at her apartment, his name is Wyatt.
4. In "The Fuzzy Boots Corollary" it is said that Sheldon has asthma and doesn't like cats and Leonard likes cats which we find in the future is the opposite of the truth.
5. Sheldon says that Amy doesn't like to wear costumes in "The Justice League Recombination", though she does dress as a pretzel, a Starfleet medical officer, Snow White and more in various episodes.
6. In "The Thanksgiving Decoupling" Sheldon says that his father died when he was 14 years old. However, in "The Pancake Batter Anomaly" Sheldon states that he was once sick in Germany when he was 15 years old, and his mother wasn't able to come to take care of him because she needed to help his father as the house had been damaged in a tornado.
Note: Checkout the Goofs sections for TBBT episodes on the Internet Movie Database and the Contradictions page in this wikia.