Specifically, in The Handmaid's Tale, Offred (the title character) is not supposed to read because totalitarian governments normally control all information that the commoners have access to. Reading is particularly dangerous in Offred's society because the glamorous magazines with their glossy pictures and sensational articles give her knowledge of what life used to be like before handmaids such as she existed. Her only role in life is to bear a child to the commander, but strangely enough, the commander is the one who allows her to read.
This concept of forced or encouraged illiteracy among the population is not unrealistic or unique under a totalitarian regime. In today's North Korea, no one (except the elite few highest up in the government) has access to the Internet. The books that they do have access to are tightly monitored and authored. In Afghanistan under the Taliban's rule, the Talibs began to encourage illiteracy because they knew thatilliteratehumans areeasier to brainwash and control.Similarly, tourists can still visit the plaza in Berlin where thousands of books by Jews or related to non-Nazi-sanctioned ideas were burned under Hitler's rule.
Other literary works that portray reading as dangerous include Bradbury's Fahrenheit 451 and Orwell's 1984 (newspapers and other materials are censored or destroyed). Because reading encourages thought, it is extremely dangerous in a regime that wants to stifle thinking and encourage blind obedience.
The book takes place under the rule of a total dictatorship in a future fictional country within the borders of the present day United States. The fact that the story is about a dictatorship is the key to your question. In dictatorships where absolute power rests with a single person or party, reading is dangerous because it gives people ideas and promotes independent thought. This is why you often see dictators taking total control over all media - what is taught in schools, the TV and radio stations, internet etc.
As Soviet leader Josef Stalin once said: "Ideas are more powerful than guns. We would not let our enemies have guns, why should we let them have ideas?"