During the ring contest both Sheldon and Howard change which hand is holding the ring.
As the four contestants arrive at the door to Leonard and Sheldon's apartment, Leonard drops the keys on the floor. When they all stoop so Leonard can reach them, the keys are in a different location.
When we see Sheldon with his plastered nose and black eyes after Penny hits him, Howard asks 'So Sheldon, how's it feel to be beaten up by a girl?' but in 'The Killer Robot Instability', Penny punched Howard after he tried to kiss her and he ended up with the same injury.
When Howard shows off the ring at dinner, after it's been verified as an actual movie prop, the position of his fingers and the chain change between shots.
While discussing the ring during dinner at a restaurant, the ketchup bottle on the table rotates. Sometimes the "57" is visible and sometimes it isn't.
The original rings worn in the movies did not have the elvish writing on them. That was added digitally in post-production.
This same type of error occurs in many episodes. When talking with Raj's cousin over the computer, his cousin looks up and to the right to talk to Raj and tell him to "Call your mother." Since the computer has a web camera at the top of the computer, the only way for the cousin to see Raj on his computer is to look straight ahead into the camera, not up and to the right like he is actually speaking with Raj in person.
In the dinner scene where they find out the ring is real Raj goes to get some tomato sauce. When he sits down his burger is slightly toppled over his chips then it cuts to Leonard and when it cuts back to Raj, his burger is standing straight. It then alternates between straight and toppled several times.
Sheldon talks about how if he had reabsorbed his sister in utero he would have a mole with hair instead of a tedious Christmas letter. Sheldon and his sister are fraternal twins and came from separate eggs, so he could not have reabsorbed her.