Most cartoons, from Tom & Jerry to the Simpsons, list a director credit- with live actors, I understand what a director does. But what does a director do with no live actors?
The guys who do the voices would not appreciate your assertion that a cartoon has no actors. 
In addition to directing the voice talent, a cartoon director is responsible for deciding the visual layout of each scene and how the characters will move around. The artists and animators then make it look how the director wants.
(probably inaccurate, from memory, from Joe Adamson’s Tex Avery: King of Cartoons.
I know that Chuck Jones used to sketch out the main layouts and key frames for a cartoon before turning it over to “inbetweeners” to do the cels in between. He also did storyboarding and would discuss the cartoon with the voice actors.
It can be very interesting to listen to the commentaries on the “Looney Toons Golden Collection” DVDs to learn about the different styles of directors. Jones was controlling, drawing all the main points that he wanted, but expected his team of elite animators to use their own skills to fill in what got to those points. Robert Clampett had a much wilder style, and gave his animators more latitude than most of the other directors. It’s really a fascinating job, and incredibly difficult.
Who is responsible for the pacing of the action? Is that part of the director’s job? I always thought that was one of the great strengths of the classic Looney Tunes.
RR
Definitely part of the Director’s job, although of course it falls to a bunch of people to make his plan work. Jones, for example, had a rule about how may frames of film it took for the coyote to fall into the canyon, and it took a damn good animator to make it come out exactly right.
Not sure about other studios, but I know that at Pixar, the Director is the final arbitrator of the movie’s story/plot/themes/style/whatever. That’s why they always emphasize that their films are director-driven, because the director is the person who has to nurture the whole movie for the three or four years it takes to complete it.