When Collier is found sitting motionless in his room, he has been there for "days" after listlessly wandering about for some time. Nevertheless, he remains clean-shaven. We know he has a beard, because he has struggled to shave with a straight razor.
When Collier walks into the Hall of History he is carrying a book. The camera angle changes and he no longer holding a book. He runs to see Arthur and still no book, but when he's eating his diner, he's shown reading the same book.
The hotel register that Richard finds in the attic in 1980 is different than the register he signs at the hotel in 1912. Notice the signatures and placement of "Chicago" - completely different.
The chain on Richard's watch when he checks the time after Arthur leaves his hotel room. It goes from being attached to Richard's pants to being in his hand.
When Richard is sitting in the garden having breakfast (before he actually checks in to the hotel), the table setting alternates between just a coffee cup and a bowl of cereal (with Richard's hat visible) and a complete breakfast (without Richard's hat).
In 1912 (and, indeed, until recently) gentlemen did not wear their hats indoors.
At the beginning of the film, Elise gives a watch to Richard after watching his first play, saying "Come Back To Me". Richard takes the watch back with him into the past, leaving it with Elise when he unwillingly returns to the present. Elise keeps it until she gives it to Richard, seen at the beginning of the film. This is a temporal paradox, since at no point in this loop does the watch actually get manufactured.
When Collier is in the library, the index of the book he looks at has several errors. The name of Cecilia Loftus is misspelled as Ceciilia. The alphabetical order is also wrong. The names are listed as May, McCracken, Merman, McKenna. The fictional McKenna may have replaced the name of stage actress Elise Murray in the book. Also the inclusion of Joan McCracken seems odd because she didn't make her Broadway debut until 1943. The other actresses named were on the stage much earlier in the century.
You haven't been able to drive on Mackinac island for over 100 years. It would have been impossible for Richard to happen upon the hotel while driving in 1980. However, while the movie is largely filmed on Mackinac Island, it's not set there. The "Grand Hotel" in the movie is located in a city where cars are allowed and is apparently somewhat closer to Chicago than the real life hotel considering Richard drives by it on a lark.
Recurring throughout the movie and specifically referred to in the lovers' conversation is the lyric melody from Rachmaninov "Rhapsody on a Theme by Paganini". This work did not exist in 1912. It was composed and premiered in 1934.
This is done by design. When Richard is taking Elise for a boat ride, he is humming it. She says that she is familiar with Rachmaninov but not this piece. That makes sense. She wouldn't know it if it was composed later, but Richard would know it because he was born after it was composed. That's why he tells her that he'll introduce it to her sometime. He knows he'll have to wait until it is composed.
In 1980, Elise could have said to Richard "Empty your pockets of everything before you come back," as she knew how she had lost him. Update: But that's assuming she understood what happened. She sees him playing around with his suit then he vanishes which was quite a traumatic moment. Considering she didn't know he was a time traveler from the future, it's actually amazing she was able to figure out that he was and track him down.
When Richard is about to cross the street carrying the box of clothing, the camera wobbles just at the end of the tracking movement, suggesting the operator hit the end of the track just slightly too hard.
When Richard checks into the hotel in the 1970's you will notice the bright floral carpet in the lobby. When he returns to the lobby in 1912 and Arthur is playing ball, you can see the lobby has brown carpet with rugs laid across it. Arthur sits on his ball and if you look closely, the brown carpeting does not go all the way to the wall, rendering a strip of bright floral carpeting visible to the eye just along the wall.
When Richard is sitting on the porch of the hotel with his back to the lawn, Elise approaches the building and screams his name. The next shot - from the lawn - shows that there is no way Elise could have seen him from where she was.
When Richard drives off into town to go to the library, the windshield wipers on his car are activated despite it being a beautiful sunny day. Apparently, in his haste to jump into the car during filming, Christopher Reeve hit the lever and turned on the wipers. The director saw this and left it in the movie to show that Richard Collier was very excited.
Near the end of the movie, as Collier has been struggling for hours to return to 1912, he drops the 1979 penny that brought him back to the current time, and would do the same immediately if he were able to return to 1912. He should have thrown it as far from himself as possible as soon as he found the cursed thing in his presence.
An American flag flying from the Grand Hotel in 1912 has 50 stars. In 1912 it should have 46 stars (early in the year), or 48 stars (after 4 July).
The song "You Made Me Love You" was first published in 1913, however when Richard first awakens in 1912 and finds himself in a stranger's room, the occupant of the room can be heard singing this song.
When Richard initially goes backstage to look for Elise during the afternoon rehearsal, he passes a hanging bare light bulb. It is a frosted light bulb of modern design (since the '40s) - not the hand-blown clear glass bulb that was in use in 1912.
The Portrait of Elise seen by Richard is a glossy gelatin print. Glossy gelatin print stock was not commercially available until the 1920s. Before this date, only mat and gloss albumen or mat gelatin processes were available.
When Richard kneels down in the hotel lobby to return the ball to the little boy (that in modern time is the bellboy), there are modern day fire sprinkler head covers in the ceiling behind him. Also, the ball is a modern inflatable playground ball.
Richard sits down on the bench near the stairs and Elise comes into view behind him, looks up, sees him and screams his name. Immediately before this, when he first sits down, a woman with short hair wearing a knee-length white flowing skirt can be seen walking in the same area where Elise comes out from behind the trees and bushes.
In the book of biographies Richard reads at the library it states that Elise McKenna was born on November 11, 1885. Later on in the film, Robinson states that he's been working with Elise since she was sixteen-years-old in March of 1903. Either she was actually seventeen or she was born in 1886 rather than 1885. So, either Robinson was misremembering or the biographer screwed up/was unable to find accurate documents.