Where should I start watching The Dark Knight franchise?

You should start with Batman Begins, which is the first movie in the release order and establishes the main character and his mission in Gotham City.

Is release order required for this franchise?

Yes, release order is required because the movies form a continuous narrative where each story follows the direct consequences and plot events of the previous entry.

Which entries can be skipped in The Dark Knight franchise?

None of the entries can be skipped, as all three movies are necessary to follow the complete storyline involving Batman, Gotham City, and the recurring characters.