When Gisella is in front of the board in 1.034 and removes her gloves the ID number on her arm is missing.
When the Russians liberate the camp, it looks like a summer evening, or at least somewhere at the end of spring time. It doesn't look like winter at all. There is no snow visible and soldiers don't have any protective clothing. Auschwitz was liberated on 27 January 1945 in midwinter Poland.
Neither Cluj nor Sighet belonged to Hungary in the period the movie refers to - except for the military occupation of Horthy's Hungary (1940-1944). Even during that brief occupation, most inhabitants of other than Hungarian ethnicity would not have identified themselves as "Hungarian". This is probably not such an innocent mistake, since that is still a sensitive issue, Hungarians still claiming that part of Romania for themselves...
Dr Josef Mengele was an SS officer, but the movie portrays him in the black Gestapo uniform.
When Gisella Perl is shopping for produce in the market she is called a killer by a women who is missing a bottom tooth. You can easily tell that the tooth is not really missing but obviously blacked out due to the reflection.
At 8:48 into the film, Gisella's father is seen in close-up, and it's obvious he's wearing a fake mustache. On the character's right side (the viewer's left) the spirit gum used to hold it in place is shiny, and the mesh backing for the mustache is visible.