Anachronisms: When everyone is skating, all of them are skating on modern hockey blades. Some of the people can be seen with white plastic blade attachments.
Errors in geography: When Heinrich Harrer uses a globe to explain time zones to the young Dalai Lama, he spins the globe the wrong way (clockwise - from East to West), and says "So if the sun is just rising in Lhasa, that means it is just setting in New York City, perhaps." But the Earth spins counterclockwise, and Harrer should have spun the globe the other way, saying "So if the sun is already setting in Lhasa, that means it is just rising in New York City, perhaps." Another possible correct statement would be to say "So if the sun is just rising in Lhasa, that means it is just setting in New York City yesterday, perhaps."
Anachronisms: Negotiators from the Peoples Republic are shown arriving by air. There was no air link until Lhasa Gonggar Airport was constructed in 1956.