Everybody Remembers the Stripper: You might think that more modern challenges to the book would involve the constant use of the word "retard" note today, really uncomfortable, but in the 50s, that was the term used or some of the unintended implications about disabled people note the story unintentionally implies that teasing and being incapable of realizing it is a natural part of being mentally disabled - totally mainstream thought in 1958, but questionable today, to say the least. But even in The New '10s, the biggest, practically only, reason the book is challenged for removal from school libraries and elsewhere? Its (utterly mild) sexual content.
Family-Unfriendly Aesop: If you were born mentally-handicapped, there is nothing you can do to change it. Neither your own efforts nor advanced medical technologies will help you. Sadly this was essentially the mindset during the time period.
Nightmare Fuel: In the 1968 movie adaptation, Charlie's mental breakdown which involves him running from his older self. The music and the creepy smile of the other Charlie are unsettling enough, but then Charlie gets trapped in a maze and at one point, he's struggling to escape down a long hallway while the other Charlie slowly approaches him. That whole scene (starting from 2:35 and going to 6:00) is beyond creepy.
Some Anvils Need to Be Dropped: The book is very clear that the treatment of the mentally disabled by abled people is horrific. Things actually have improved in the treatment of the developmentally disabled since then - and probably at least partially as a result - In the 1950's it was very common for families to send away these folks and they were often encouraged to do so. Nowadays its more acceptable to have a special needs child.
The Woobie: Charlie, obviously. Realizes that his so-called friends were simply using him as a toy, finally reaches genius levels of intelligence, only to lose it, and starts to forget almost all the notable events that happened to him towards the end.