The Day of Black Sun (1): The Invasion - S3-E10
Corrected entry: Ming brings Iroh White Jade Tea, though Iroh was poisoned by it in S2-E2: The Cave of Two Lovers.
Character mistake: When Jet tells Katara and Aang to waterbend the water from underground in the geysers, Katara states that she has never waterbent on water she couldn't see before. However, six episodes prior to this, in 'The Warriors of Kyoshi', she waterbends the water out from Aang's lungs to save his life.
Suggested correction: That was a moment of desperation on her part, and she probably doesn't think of it/think she can do it again without the same desperate impetus, Furthermore, the water in Aang was only a small amount and very close anyway.
Regardless of if she did it in desperation or not, she did use bending on water she didn't see. The mistake isn't suggesting anything about Katara's ability to waterbend unseen water, the mistake is only in regards to the statement Katara made about never using bending on water she can't see. And for Katara to forget the one time she used bending on water she didn't see would qualify as a character mistake as well. Now if she made the statement "I don't think I can do this" or "I've never used bending on underground water", etc, that would not be a character mistake.
The Boy in the Iceberg - S1-E1
Continuity mistake: In this episode, Katara remarks that "This ship has haunted my tribe since Gran Gran was a little girl. " However, in the episode "The Puppetmaster" it is revealed the ship wasn't beached there until Gran Gran was in her late teens or early twenties.
Suggested correction: You sure that was the same ship, and what's 5-10 years give or take compared to Gran-Gran's life anyway?
Correction: Actually, during The Cave of Two Lovers, he said the flower was either the white jade or a poisonous flower that would kill him. He apparently decided against the tea and found some berries, that he thought was safe but ended up not being. The berries are what almost killed him.