According to the timeline of the movie, a large part of this would've taken place during late fall to winter months. (Plans to visit Ray's parents for thanksgiving, Scott talking to his sister at the Halloween party, etc.) However, the weather is summer like the entire movie.
While she's at the college party, Claire's clothes change.
Scott's gloves switch between on and off after giving Harold his tattoo.
In the emergency dept., the blood on Ray's shirt is inconsistent between shots.
When Scott quizzes Kelsey at the end of the movie, the answer given to his first question is incorrect, at least as worded in the scene. If '"Amy must complete three tasks at work" and the first two take "45 minutes," while the third will take "two hours and 10 minutes," it would take her two hours and 55 minutes (45 minutes+2 hours 10 minutes=2 hours 55 minutes) to complete all three tasks, not "three hours and 40 minutes," as Kelsey answers. It is only if the first two tasks took 45 minutes EACH, that the correct answer would be three hours 40 minutes. It is possible that either there was an editing error, or that the original source test was poorly worded.
At roughly the half way mark, when Scott says he's "just kidding" about "training" Ray's daughter in the car, you can see his lips aren't moving in the shot behind him.
When Scott has breakfast at home with Ray for the first time, Ray talks about different celebrities' accomplishments by age 24. Ray says that Roger Stabauch graduated from West Point. Roger actually graduated from The Naval Academy 1963.