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 printing out paper was not commercially available until the 1920s. Before this date, only mat and gloss albumen or mat gelatin processes were available.
When Richard follows Robinson and Elise from the lake to the hotel after they first meet, the road they walk along is briefly shown and it is modern asphalt pavement with dotted white centerlines. Centerlines (solid, not dotted) did not begin to appear on rural US roads until several years later during WWI and would not become commonplace for many years after that.
Most roads at this time were unpaved, cobbled or lined with crushed rocks, such as gravel. What few miles of paved rural roads there were would have most likely been "macadam" or "tarmacadam" type roads commonly in disrepair. Layered asphalt roads similar to today's roads was in its infancy and years from widespread use and the maintenance infrastructure that is needed to maintain them.
Although Rachmaninov's work "Rhapsody on a Theme by Paganini" did not exist in 1912, it did exist when Richard Collier was alive in 1980. When Elise and Richard are rowing together on the lake and he is humming it, Elise mentions that, while familiar with Rachmaninov, she is not familiar with this work. Having gone back in time to 1912, Richard would be familiar with the work and Elise would not which could lend itself to Richard's comment to Elise, "I'll introduce you to it sometime," perhaps meaning that once their relationship is further along, she would accept that he would know of a piece of work that had yet to be composed.
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.
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.
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.
(at around 8 mins) Richard checks into the The Grand Hotel in 1972. The receptionist checks him in without glasses. In the next shot the receptionist is wearing glasses and hands Richard his room key. Seconds later, in the next shot the receptionist is again without glasses.
When Richard goes to the front desk to get a room, Arthur's father takes Arthur's ball away and puts it in a space behind the counter that is not visible to anyone who is in front of the counter. While Arthur's father is doing this, Richard is watching Arthur who is behind him. He does not see where Arthur's father has put the ball, and yet after Arthur's father moves away, Richard somehow. knows exactly where the ball is.
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).
Robinson tells Richard that he first met Elisa in March 1903 when she was 16 years old. However, her date of birth was given as November 11, 1885 in one of the theatrical books that Richard read earlier in the film. Consequently, she would have been 17 in March 1903.
In 1972, when the elderly Elise gives the watch to Richard, it shows the time as 11:20 and about 45 seconds. In 1980, when he checks the watch in his room at The Grand, it has moved on by about 5 seconds; clearly shot at (almost) the same time.
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.
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.
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.
The goof item below may give away important plot points.
At the end of the movie when Richard grasps the new 1979 Lincoln penny, the realization of the time paradox jerks him back to 1980. However, when he's rowing in the boat with Elise she comments on the melody he's humming and he pauses, realizing it didn't exist in 1912, and then covers by saying he'll introduce it to her some time. Why didn't this realization of the time paradox send him back to 1980? After all, it was his self hypnosis that brought him to 1912, meaning the power of his mind was holding him there in 1912. The time paradox realization of the penny was sufficient to break his mind's hold on 1912 so the music paradox should have done the same thing. Similarly, when he tells the clerk the room he's supposed to get and the correct time of his data entry (Bingo!) as he remembered from seeing it in the old register in 1980, that also demonstrates the realization of a time paradox. When he tells Arthur, "See you around Arthur" and Arthur asks if they've met before Richard lies "No". Again, time paradox realization. When he first meets Elise at the lake, he tells her he's come so far to be with her. Time paradox realization again. When Richard tells Elise on their afternoon walk that the watch was given to him as explanation of where he got it, he was definitely thinking of the time paradox. Without watching the film again and looking for more evidence like this (I always get too into the movie to ever do that) I'm sure there would be other incidents like these. I suppose all Richard's mental hiccups could be dismissed for not sending him back to the future (another great movie) as they were only thoughts, but the physical reality of the penny was sufficient, even though the physical penny created a thought in his mind, breaking his hold on 1912. This was demonstrated by Richard's initial failed attempt when he realized the physical presence of the cassette tape recorder prevented his mind from taking him to 1912. I guess objects are more powerful than ideas in cases of hypnotic time travel.