These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Harsher in Hindsight: Chaplin only had a faint picture of the atrocities occurring in Nazi Germany when the film was being produced, based on what refugees from Europe had told him. Back in 1940, the worst things have not started yet anyway. He stated that had he known the full extent of the Nazis' crimes, he'd have never made the film.
The concentration camp scenes in particular make for quite awkward viewing. Perhaps the most In Your Face example of this trope is when the Jewish Barber first arrives at the camp. Whereas all the prisoners walk in one direction, he walks toward another. When one of the guards asks him where he's going, he cheerfully answers: "The Smoking Room," before being pushed the other way. In Auschwitz, prisoners were pointed to walk in one direction or another, and people waved in the wrong direction got sent to gas chambers.
Hynkel makes a stirring speech about Tomania's expansion, including France, Finland and Russia. France had already been invaded by Nazi Germany, Russia would be in the next year, and Finland would remain Germany's co-belligerent during the latter.
Heartwarming In Hindsight: With the allegations Nazi Germany's influence on Hollywood during its golden age coming to light, Chaplin's determination to produce an anti-Nazi satire feature film in defiance of it comes off as downright heroic.
Memetic Mutation: For some reason the Rousing Speech has been seeing some spread, in the form of clips like this one. Such clips are rather frequently posted under the title "The Greatest Speech Ever Made", or some variant of it.
Chaplin says he would've given anything to know what Hitler thought.
Reality Subtext: The film was prompted when someone pointed out the visual similarities between Chaplin's "Tramp" character and Adolf Hitler.
Some Anvils Need to Be Dropped: The anti-nazi message isn't exactly subtle but this was the US of 1940 and a large minority in the US had some sympathy for or were impressed by the nazi regime. Effective, easy-to-understand and revealing satire of nazism and fascism was badly needed.
Stock Phrase: "Schtonk!" alone stands for the whole film (up to the point that the German film parody of the fake Hitler diary affair is called this way).
What Could Have Been: In his autobiography Chaplin, who'd resisted for ten years going over to talkies, revealed that his original concept of the film was to mix silent sequences featuring the Tramp with talking sequences featuring The Great Dictator.