In the song "I'm Reviewing The Situation," Fagin imagines himself owning "a suite at Claridge's." However "Oliver!" is based on Dickens' "Oliver Twist," which was set in the 1830s, and Claridge's Hotel was not founded until 1854.
In "Who will Buy", just after the school girls are pushed into the pool, a family leaves a house and the mother is wearing hat and clothing more appropriate to the 1890s rather than the 1830s. She has on a large hat with a brim as opposed to a bonnet and her dress is definitively from a later era.
There is a pub in the main street called "The Victoria", and the post box Dodger hides behind at the end of the film is embossed with "VR" for "Victoria Regina". Oliver Twist is set in the 1830s, and was published 4 months before Victoria's accession.
Nancy, Bet, their fellow barmaids and the saleswomen of WHO WILL BUY all wear their hair loose in what was one of the most popular fashions of the late 1960s; the high class ladies of the latter number, on the other hand, wear their hair in the correct fashions of the 1830s.
Mrs Bumble is seen ringing the bell hanging from the ceiling at beginning of the film telling that breakfast is ready. But when the boys come down the stairs singing "Food, glorious food", her and the bell disappear.
During "who will buy" the school girls are pushed into the pool in the park by the school boys. The next scene shows the school girls with dresses that are all dry and pressed instead of wet and droopy as they should be.
The "Boy For Sale" scene is historic nonsense: even during the harshest phases of the Poor Laws, a workhouse pauper could not be traded like a slave from one "owner" to another. Apprentices could be had for nothing, and it was state representatives like Bumble who had to pay tradespeople like Sowerberry for the service of taking paupers off the state's hands, rather than the other way around. The book got this right.
The steps at London Bridge are depicted wrong. The real steps were narrow at the top, and widened horizontally at the bottom, allowing a person to hide and eavesdrop on (or watch) those on the upper steps without being seen. The book got this right.
In the number, "Who Will Buy", the question is asked, "Where is the man with all the money, it's cheap at half the price". Anything is cheaper at half the price. The correct statement would be, "It's cheap at twice the price," which is the original English adage.
When Oliver starts his journey from the workhouse town to London, it's winter. As his travel proceeds, the winter snow melts, and the landscape becomes green. By the time he arrives in London, it's clearly summer, judging from the people's clothes and the abundance of vegetables. While conversing with the Artful Dodger, Oliver explains that he has been walking for seven days.
Just after Oliver asks for more gruel and is taken by Mr. Bumble to the governor of the workhouse, they are standing at the door - Oliver mouths Mr. Bumble's lines, and then to cover it up, starts wiggling his tongue.
The goof item below may give away important plot points.
At the end, Fagin and Dodger walk out of the city down a long cobblestone street. As they walk directly towards the sun rising in the distance, with their faint shadows seen following them appropriately, we also see larger, more pronounced shadows on the brick wall to their left - which is at a full 90 degree angle away from the sun in front of them.