I found this nice tutorial that explains how to convert a movie file to a .gif in Photoshop. Is there an analogous process in Illustrator?
Illustrator has no animation set up specifically. There's no structure to interpret existing animation or video within Illustrator.
In fact, the only places Illustrator even remotely addresses any sort of animation is in the ability to build layers in a sequence, and to export to swf format (Flash).
Illustrator can not interpret a video file in any way. Therefore an .mp4 would mean nothing to Illustrator. Illustrator won't even recognize an mp4 file.
Illustrator is also incapable of building gif animations. Even starting from scratch, there is no method to build a timeline or frame animation within Illustrator.
There's nothing you can do with an mp4 in Illustrator and you can not create animated gif files with Illustrator either.
In short... No. There's no analogous processes with Illustrator.