Moments after Tyler accidentally causes the mess in the office pool before he enters Drysdale's office, it is exactly the way it was before the mayhem.
When Tiffany brings Jethro more pencils to sharpen, a bundle of unsharpened pencils disappears in between shots.
When Miss Hathaway goes to see the private agent, his name plate on the desk disappears then reappears in between shots of Hathaway and Jones.
Elly May was barefoot when harvesting honey in the opening scene. During her fight with a bear during that time, she was wearing shoes. After the bear fight, she was barefoot again.
Derek is doubled-over on the mat after Elly May delivers the "hickory nut crunch", but he's up in the next shot, ready to wrestle.
Some of the puppies that Mrs Drysdale's dog has are bloodhounds and some are poodles. They should be mixed breeds that have characteristics of both parents.
When "Jed was shooting at some food" and fired the shotgun, there was a noticeable delay of about a sec between the shot/puff of smoke and his jerking his shoulder back due to the recoil on the shotgun. If the gun had really been loaded and there'd really been that much recoil, his shoulder would have moved at the same time we saw the smoke/flash of the gun.
Mr Drysdale tells Tyler that The Clampets are not the run of the mill everyday garden variety millionaires. They are in-fact billionaires. This would actually be inaccurate as He's Clampet was given 1 billion dollars for his oil. Minus the cost of how much he must have paid for their mansion, moving costs and whatever else they may have purchased or done with their money, they would be well under a billion dollars. They would be multi-millionaires, but not billionaires.
The lunch cart in the office disaster scene bounces after the screaming lady falls over it.
When Jethro is lying on Miss Hathaway's computer monitor, Hathaway is obviously typing vigorously but when the camera pans to Jethro, we see nothing is being added/changed on the monitor
When the Drysdale's dog has a littler of pups under one of the tables, the puppies are obviously 4-5 weeks old as at least one of them is up and walking around.
During the scene where the whole family is first driving into Beverly Hills, in the scene where they wave hello, the camera crew can be seen in the reflection of some of the cars.
From Pearl's talk with Jed after he's struck it rich, it's clear that the Clampetts are ignorant of modern conveniences, yet Granny later makes reference to a telephone as though she has used one her entire life.