Just for starters on opening day:
There were so many people, the restaurants ran out of food and beverages.
Due to a plumber's strike, they had to choose between working bathrooms and working water fountains. They chose bathrooms.
There was a gas leak in Fantasyland.
Many rides broke down. The Storybook Land Canal Boats had to be pulled by a cast member in rubber boots. There were no guard rails at Autopia so many cars crashed and most became completely inoperable.
The asphalt on Main Street had just been poured the night before and was still wet, so women's heels sank into the concrete.
Within the first few weeks, the stagecoach ride had to be discontinued because it kept flipping over, a tiger and a panther got loose during a circus parade and got into a tussle on Main Street and the Mark Twain almost sand and got stuck in the mud because too many people were onboard.