During the opening credits, a scene shows the Train passing a boat on the river. The boat clearly has an exhaust funnel and is pumping out bilge water. The film is set in 1890, and diesel engines for boats didn't exist until 1897, and electric bilge pumps weren't common on many boats until the 1940s (the were more commonly used on large naval/merchant vessels).
During the number "Put on Your Sunday Clothes" - a few minutes into the number - Dolly, Emergarde and Ambrose do a high step down a flight of stairs in front of Vandengelder's Hay and Feed. The camera changes angle and across the bay, where there is a highway, there are modern era (1969) cars and trucks whizzing by.
During the opening sequence: When "Based On The Stage Play 'Hello Dolly' Produced by David Merrick and Book of Stage Play by Michael Stewart" comes on the screen, you can see in the background an interstate highway going around the base of the tall hill with cars whizzing by.
During the opening still, there is a title card stating "New York City 1890". In the background is a horse-drawn Bekins moving van. According to the Bekins website, they weren't founded until 1891 in Des Moines, Iowa.
During the "Hello Dolly" song at Harmonia Gardens towards the end of the movie, she's singing with Louis Armstrong, after they do their exaggerated "yeah..." she's supposed to sing "Dolly will never go away..." Her voice sings the word "will" in that phrase, but her mouth does not.
In the park during "It Only Takes a Moment," the group of bystanders starts to sing. There are only 9 people, and only 3 of them women, yet the chorus heard is clearly much bigger and much more female.
During the dancing lesson that Dolly gives Barnaby, Irene and Minnie return from changing their clothes and Minnie Fay's dress alternates repeatedly from full-length to three-quarter-length until they all dance out the door.
The red-carpeted staircase at the Harmonia Gardens restaurant has brass carpet rods on each stair during all of the scenes prior to the arrival of Dolly Levi. When she arrives and they sing "Hello, Dolly," the carpet rods are gone.
When Horace is marching in the parade, Dolly approaches him to talk so he holds the sword upward on his right shoulder. Between cuts, when she talks about dinner at Harmonia Gardens, the sword disappears repeatedly.
In Harmonia Gardens, after Cornelius says that he is "Gonna become a honest man and tell the truth," he puts the bottle on the table and goes to sit near Irene, holding the drinking cup. When he sits down the drinking cup has disappeared.
In the chase sequence in the Harmonia Gardens restaurant, Horace Vandergelder is seen entering the fireman's pole in an anticlockwise direction. When he exits at the bottom he does so in a clockwise direction.
After Horace leaves for New York, Dolly is wandering around in the store. Just before she walks out of the door, she slings her purse over her right shoulder and holds it with her right hand, but when the angle cuts to her coming out of the door, she is holding her purse with both hands down in front of her.
When Dolly comes down stairs of Harmonia Gardens, her dress has a train that trails several steps behind her, but by time she reaches bottom, train has disappeared - a mid-scene alteration allegedly made necessary by Streisand's difficulty in dancing with the trailing gown.
When Horace joins Dolly at her table, she spoons a great deal of food onto his plate, including lots of beets, but when the camera angle changes to behind Horace, there is very little food on his plate and no beets are there.
During most of the song "It Takes a Woman", one man is standing behind the horse and the remainder of them are either to the right of the horse (from the audience's perspective) or directly in front of it. Toward the end of the song the camera angle changes and suddenly there are several men to the left of the horse.