We see the new moon very close to the sun (though showing far too much crescent for such a small angle). A few scenes later, Brian Marsh is looking at a full moon, then a few more scenes later, we see the new moon again near the sun. That night, we see the full moon again.
Toward the end when Brian jumps out the window and the possessed people outside come towards him, it is clearly dawn and light outside. Shortly after, when the possessed is defeated and the mirror is broken by the priest, you can see them dispersing while it is night outside. Then Walter jumps out to get help and it is night as well. The next scene shows Walter with a blanket around him and is dawn again.
(at around 38 mins) The student radiologist Susan Cabot says she has been "carbon dating the corrosion on the lid" of the cylinder, and "we're talking about 7 million years". Carbon dating is used to determine the age of materials, usually organic, containing carbon, not metal corrosion substances, and is only useful for about 60,000 years.
(at around 12 mins) In the beginning of the movie the Priest tells Dr. Birack that the Catholic Church containing the demonic fluid, located in Los Angeles, was built in the 1500s by Spanish missionaries. The Spanish did not colonize California with missions until 1769, with the founding of San Diego by the explorer-missionary Fray Junipero Serra. The mission where filming took place, Mission San Fernando Rey de España, was not founded until 1797.
(at around 1h 29 mins) After Walter crawls through the hole in the closet with "possessed" Lisa in tow, they're both in the other room with Prof. Birak, all standing up. You can see the obvious doubling of "short Lisa" by a stuntman - she's suddenly much bigger and taller than the two other men.