(at around 19 mins) Pamela Voorhees' tombstone reads "1930-1979", but Friday the 13th (1980) shows her death taking place on Friday, June 13th, and June 13, 1979 was a Wednesday. (The year of her death should be 1980, as the first film was released in that year, and June 13th indeed fell on a Friday that year.). The Part 2 trailer and 2009 remake confirm that Part 1 did indeed take place in 1980.
(at around 1h 23 mins) Jason crushes a lamp on the floor and for an instant it becomes pitch black. A moment later, the room is lit again by a sudden occurrence of "moonlight."
In Friday the 13th: Part 3 (1982), Jason's fingernails were neat and trimmed, but in 'Part IV' (which takes place the very next day), they are long and black.
(at around 15 mins) When Tommy is playing Zaxxon (1982) on his computer and talking to his mother, if you watch closely his ship is about to hit a force-field and he is going to die in the game. They cut away and a second later he is nowhere near the force-field with no chance of dying in the game.
(at around 15 mins) Tommy isn't wearing glasses while he is playing his computer game, so it can be assumed that he needs them for distance. However, later that night, when he is watching Paul and Sam through the window (at around 24 mins), he doesn't have his glasses on but he can still see what is going on.
(at around 32 mins) When Trisha's car breaks down the solenoid is cited as the cause. A solenoid controls voltage to the starter (which was working) and would not cause a vehicle's engine to stall.
At the beginning of the movie (after the flashbacks from the previous movies), there is a police helicopter flying above the crime scene. There was actually no need to deploy a helicopter because the scene was contained, and they were not doing a manhunt.
(at around 1h 17 mins) When Trish shouts, "Tommy, hurry", the shelf to her right gets bumped by a crew member.
At 12:15 when Jason's arm falls out of the blanket in hospital, you can see a tribal tattoo on his forearm.
On the cover of the 'Friday the 13th' case, the ax mark left from Friday the 13th: Part 3 (1982) is located on the left side (looking at the case), while in that movie and this one, the mask mark is located on the opposite side.
(at around 14 mins) When Trish and her mom are jogging, a bush in front of them moves before they run by it, caused by the cameraman brushing it as he walked backwards.
In Friday the 13th: Part 3 (1982), Jason was stabbed in the leg and his hand, yet in 'Part IV' (which takes place the next day), there is no blood stain nor cut in his trousers or his leg, and no wound on his hand.
Friday the 13th Part 2 to 4 are set over one weekend, Yet Rob has time to find out his sister was murdered find out Jason did it and track him down.
The Jarvis' car stalls. Ron fixes it and says it was the starter. The car was already running and stalled out, which means that the issue could not have been the starter.