When Nigel is recruiting the Marmosets for help, he threatens three monkeys after he tosses the main one in the sky. One monkey has his hands over his eyes, another over his ears, and another over his mouth. (See, Hear, Speak No Evil).
Carlos Saldanha (himself a resident of Rio) first came up with the concept in 1995. Originally, it was about a penguin washing up on the beaches of the Brazilian city but when he learned of the production of two penguin-related animation features - Happy Feet (2006) and Surf's Up (2007) - he was forced to radically rewrite his screenplay.
Right before Blu and Jewel's escape from the smugglers hideout it is possible to see that the soccer game on TV is a match between traditional rivals Brazil and Argentina. This plausibly explains why everybody seems to be so obsessed in following the game - especially because the movie seems to indicate that the match is being held in the Maracanã Stadium, in Rio.
During production, Jesse Eisenberg was still working on The Social Network (2010). To compensate for time, he agreed to provide Blu's dialogue on weekends, admitting that it diverted him away from the mindset of his nearly-joyless Social Network character.
When Blu tells Jewel to check his math to verify that his plan to break their chain will work, a patch of sand is shown where Blu has scratched in the equation "f=G (m1 x m2 / R^2)". This is the formula for calculating the gravitational attraction between two bodies and can apply both literally to the rock and chain/Earth, as well as figuratively to Blu and Jewel.
The film lost out at the Academy Awards for Best Original Song to The Muppets (2011). This was the first time only 2 songs were nominated in the category instead of the usual 3 or 5. This lack of nominees in this particular category attracted widespread criticism and led to a full slate of 5 nominees the following year.
During Rafael's first on-screen appearance, he starts addressing a few of his children by their names Manoela and Sofia. These names happen to belong to movie director Carlos Saldanha's real-life kids.
The trivia items below may give away important plot points.
When Tulio dresses as a Blue Macaw, he looks himself in a mirror and says "Cyanopsitta Spixii", referring at the species Blu and Jewel belong. Cyanopsitta Spixii actually is a critically endangered bird, and its looks are a lot like Jewel. There are only 70 specimens kept on zoos and ornithological collections.
The movie has several elements similar to Happy Feet (ex. both feature singing and dancing birds, the movie ending with the main male and female characters' offspring being seen, and some rude humor early on).