(at around 3 mins) When Po wakes up in the beginning of the movie, a throwing star falls down and lands on the floor. Later, when he tries to throw it back up on the wall, the original throwing star is still in the wall.
(at around 36 mins) When Master Shifu throws two clay discs into the air for Master Tigress to demonstrate the proper use of a split kick, you can see more on the ground. The aerial view is missing the other discs.
(at around 58 mins) Before Po catches the last dumpling, he jumps over and behind Shifu. In the following close shot, Shifu is looking straight up as if Po was still right above him. Moreover, in this close shot, Po's shadow passes over Shifu. This is inconsistent too, as Po's shadow clearly points away from Shifu in the next shot. As Po was already behind Shifu, so was his shadow.
Oogway says: "Look at this tree, Shifu. I cannot make it blossom when it suits me nor make it bear fruit before its time." However, the peach tree is both blossoming AND bearing fruit which is impossible for a peach tree. They blossom first and then bear fruit much later in the season.
Tai Lung, a snow leopard, is at times heard emitting a tiger-like roar - a sound snow leopards are unable to make due to their lack of a larynx. Since this is a fantasy where snow leopards and other animals talk, wear clothing, and have other anthropomorphic abilities, this is unremarkable.
The goof item below may give away important plot points.
When Po is fighting Tai Lung, Po hits Tai Lung right into a house destroying a part of it. Yet, after he defeats Tai Lung and the villagers come back, you can see that the house is clearly fixed as if it never happened.