The Jap guard's leg is grabbed by Pinhead, he falls to the ground and the top of his head is Drilled into by Tunneler, showing blood all over his face at that time, next the camera jumps to a side view and the guy has no blood on his face.
The female Japanese agent mentions Kamikaze attacks.
No Kamikaze units existed until 1944.
The USA Flag hanging on the wall in Danny's room has 50 stars on it. From February 14, 1912 through January 2, 1959 (which covers the time this movie took place) the USA only had 48 states so there would only have been 48 stars on it (1 for each state). It wasn't until August 21, 1959 when Hawaii became the 50th state that 50 stars appeared on the flag. The 4th of July after a state is admitted is when the star(s) are officially added.
The concentration camp Auschwitz is mentioned. This camp was build two years after the events depicted in this movie.
During the opening credits, you can see the marquee for a theatre. The film being shown is How to Murder Your Wife. The film was released in 1965. Puppet Master: Axis of Evil is set during World War Two.
The swastika on Max's armband is backwards.