Charlie Chaplin's career started going downhill in the 1940s for a number of reasons, mostly political. J. Edgar Hoover had long held a dislike for Chaplin and believed him to be a communist, so he used his considerable resources to launch several FBI investigations into him in order to tarnish his reputation. There was a paternity scandal involving Chaplin and a much younger actress named Joan Barry, who accused Chaplin of fathering a child with her. Hoover charged Chaplin with a number of things relating to this case, including violating a law that was called the "Mann Act" which prohibited traveling with a woman across state lines for sexual purposes. Chaplin ended up being acquitted but substantial damage had been done to his reputation. This scandal had been the largest in Hollywood in over 2 decades and Chaplin ended up fleeing the US for Switzerland. He attempted to revamp his image with a 1947 film called "Monsieur Verdoux" but the movie flopped, with many reviews condemning Chaplin. He had failed to take up US citizenship, which caused a number of people to consider him a traitor giving support to the enemy. His communist leanings and support for the Soviet Union during the war didn't help much when the Cold War and HUAC became the focal points of US politics toward the end of the decade.

For a few reasons. First, Chaplin was seen as sympathetic towards Communists which, at the time, was worse than being sympathetic to Fascists (by a wide margin). The Nazis had long believed that Chaplin was a Jew and when he replaced his "Tramp" character with a Jewish barber character, it merely confirmed to some audiences that he was Jewish. Jews were not widely accepted in America or England at the time, with many pundits and politicians equating Communism with a Jewish plot, so it was easy to assume Chaplin was a Communist Jew.

Second, the film itself was actually well-received, from what I understand. It may be that he was cutting too close to the bone with his speech and many people didn't (and still don't) get sarcasm. Or that he went from flippant, silly comedy to a serious 'lecture' may also have turned his audience off - they couldn't separate his comedy from his serious politics.

Finally, Chaplin did go on to make more films, right up to his death in his late 80's, so this wasn't a career-ending film for him, but his work and popularity did decline throughout the latter half of his life - that's just changes in taste, I think.

Beyond the really good political reasons already mentioned here, Chaplin was just kind of going out of style in 1940. Modern Times, the film he made before The Great Dictator in 1936 was, for all intents and purposes, a silent film. It has some very occasional sound effects, next to no talking and Chaplin doesn't say a word. That's pretty important since The Jazz Singer came out 9 years before Modern Times and by 1936 sound was fairly well entrenched in American film. The transition from silent to sound theaters was still going on and it was fairly common for studios to make a sound and silent version of a movie, but by 1936 nobody in America was making completely silent films like Modern Times. I also think it's worth noting that even in his heyday Chaplin was always a very old-fashioned director. His directing style is stagey and he typically liked to set his camera up in one spot and let himself and his actors do most of the work in a scene. Compared to somebody like Buster Keaton who was an incredibly talented and creative cinematographer, Chaplin's directing style just wasn't his strength.

Chaplin's first film as the Tramp came out in 1915 and Modern Times was technically the last film he made as the Tramp. Chaplin was basically on top of the film industry for over a decade and even Modern Times and The Great Dictator were received very well. However, by 1936 two of the people he founded United Artists with (Douglas Fairbanks and Mary Pickford) had retired from acting and D.W. Griffith's career was certainly on the way down. Even though he was still doing fairly well at the box office and being received well critically, by 1940 Chaplin was getting a bit too dated and old fashioned. Then his political opinions started going out of style and his next two films released in 1947 and 1952 were received fairly savagely by the US public. After Limelight he only released two more films neither of which did well, to say the least.