Tycho Brahe's birthday is Dec. 14, 1546, but when Burke and Caffrey go to the planetarium, the employee sets the sky for 1592.
When Mozzie shows Neal the warehouse cam the IP address they are connected to is 192.553.0.55 - which isn't a real IP address as numbers in the IP address notation can only go up to 255.
The "Tycho Brahe manuscript" largely consists of pages (some upside down) written in Ancient Greek in a style not consistent with Tycho Brahe's age and time. While it is debunked as a forgery later in the episode, as a forgery it is too obvious to have ever been featured in an exhibition.
At the sundial Mozzie said the time was 4.30 in the spring, but not the government forced daylight saving 4.30, the real 4.30.