The eye is just a pinhole that leaks light onto a bunch of reactive cells. It's the brain that "sees" things. And if something goes wrong in the brain, all kinds of things can appear distorted. But one particular crossed wire brings moving objects to a standstill. It's called akinetopsia.
The first documented case of this was in 1911, when a woman who had suffered a head injury complained that she couldn't see objects when they moved too fast, especially towards her. She suffered other visual problems, including problems seeing color and general confusion. The case was so singular that not much was made of it at the time. Then in 1983, another woman, who had a lesion in her striate cortex (the visual cortex), suddenly stopped being able to see motion. But it wasn't as if she was seeing the world in freeze frames. If lights or images flickered, without moving, she was able to see the change. She was also able to see the motion of her surroundings as she walked or was driven around. It was only single, rapid motions that eluded her.
With the removal of her sense of motion, the patient - called LM in the case history - complained that daily life became extremely hard. When she tried to pour herself a glass of water, the water seemed to freeze like ice until she felt it spilling over the edge of the glass. She also had a hard time judging the heat of things, since all steam coming off an object froze, and it was impossible to tell if there was a lot of steam or just a little. Social interaction was nerve-wracking, as people who had seemed to be sitting across the room would suddenly appear before her, and she was unable to judge the rapidly changing expressions on people's faces as they talked.
It's fortunate that akinetopsia is rare, since the only real progress people seem to have made on it has been inducing it in animals. Macaque monkeys that had injections in certain areas of their brain became akinetopsic, although the effect wore off after a few days. Between 1911 and 1983, scientists observed that people who had been partially blinded could sometimes see motion, even though that motion wasn't associated with a change in color, perspective, or shape. As far as we can tell, there is one part of the brain that sees motion independently of everything else. Take that part out, and people will stop being able to see what moves, even if they can see everything else.