WARNING: Do not read unless you are up-to-date with the latest chapter of the manga!
Things that was introduce into Naruto without any explanation that sometimes contradicts with the actions of a character, and/or was never mentioned again:
Orochimaru's Edo Tensei on Yondaime Hokage
When Oro summoned the 1st, 2nd, and 4th Hokage with Edo Tensei, the 3rd stopped the 4th's summoning with some kind of Jutsu canceling technique. Yet it was later explained that the 4th's summoning failed because he performed Shiki Fūjin before his death, and not because of the 3rd's Jutsu canceling technique.
Kyuubi's Bijuu Dama
It is stated that the Yondaime Hokage sealed away the "Yin" part of Kyuubi's chakra with Shiki Fūjin. If that's so, then how does Kyuubi perform the Bijuu Dama with requires a mixture of "Yin" and "Yang" chakra?
The Uchiha massacre
The main reason for the massacre was because of the coup d'etat that the Uchiha were planning, but didn't the clan chose to turn their backs on Madara when he tried to fight for the Hokage's position? So why did they suddenly decided on a coup d'etat after so many years have passed?
Nagato's Rinnegan
Tobi claims that he gave his Rinnegan to Nagato, but didn't Nagato awaken his Rinnegan at a very young age when his parents were killed?
The Six Paths of Pain
Nagato pretty much turned himself into a radio tower with dozens of chakra transmitter in his body in order to control and grant 6 corpses the power of the Rinnegan. But all Tobi needed was to stick a chakra receiver into each Jinchuuriki in order to control them, not to mention that they did not manifest any of the Rinnegan's powers.
Naruto's unique Tailed-beast-mode
Why is Naruto's tailed-beast-mode so different from all 8 other Jinchuuriki although it was clearly stated that their power are the same regardless of their number of tails.